On classification of groups generated by 3-state automata over a 2-letter alphabet

We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients...

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Datum:	2008
Hauptverfasser:	Bondarenko, I., Grigorchuk, R., Kravchenko, R., Muntyan, Y., Nekrashevych, V., Savchuk, D., Sunic, Z.
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Veröffentlicht:	Інститут прикладної математики і механіки НАН України 2008
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spelling	irk-123456789-1523892019-06-11T01:25:42Z On classification of groups generated by 3-state automata over a 2-letter alphabet Bondarenko, I. Grigorchuk, R. Kravchenko, R. Muntyan, Y. Nekrashevych, V. Savchuk, D. Sunic, Z. We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary calculations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite rational functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of order no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other examples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups). 2008 Article On classification of groups generated by 3-state automata over a 2-letter alphabet / I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekrashevych, D. Savchuk, Z. Sunic // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 1. — С. 1–163. — Бібліогр.: 50 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20E08. http://dspace.nbuv.gov.ua/handle/123456789/152389 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic information, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary calculations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite rational functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of order no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other examples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups).
format	Article
author	Bondarenko, I. Grigorchuk, R. Kravchenko, R. Muntyan, Y. Nekrashevych, V. Savchuk, D. Sunic, Z.
spellingShingle	Bondarenko, I. Grigorchuk, R. Kravchenko, R. Muntyan, Y. Nekrashevych, V. Savchuk, D. Sunic, Z. On classification of groups generated by 3-state automata over a 2-letter alphabet Algebra and Discrete Mathematics
author_facet	Bondarenko, I. Grigorchuk, R. Kravchenko, R. Muntyan, Y. Nekrashevych, V. Savchuk, D. Sunic, Z.
author_sort	Bondarenko, I.
title	On classification of groups generated by 3-state automata over a 2-letter alphabet
title_short	On classification of groups generated by 3-state automata over a 2-letter alphabet
title_full	On classification of groups generated by 3-state automata over a 2-letter alphabet
title_fullStr	On classification of groups generated by 3-state automata over a 2-letter alphabet
title_full_unstemmed	On classification of groups generated by 3-state automata over a 2-letter alphabet
title_sort	on classification of groups generated by 3-state automata over a 2-letter alphabet
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2008
url	http://dspace.nbuv.gov.ua/handle/123456789/152389
citation_txt	On classification of groups generated by 3-state automata over a 2-letter alphabet / I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekrashevych, D. Savchuk, Z. Sunic // Algebra and Discrete Mathematics. — 2008. — Vol. 7, № 1. — С. 1–163. — Бібліогр.: 50 назв. — англ.
series	Algebra and Discrete Mathematics
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fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Number 1. (2008). pp. 1 – 163 c© Journal “Algebra and Discrete Mathematics” On classification of groups generated by 3-state automata over a 2-letter alphabet Ievgen Bondarenko, Rostislav Grigorchuk, Rostyslav Kravchenko, Yevgen Muntyan, Volodymyr Nekrashevych, Dmytro Savchuk and Zoran Šunić Dedicated to V. V. Kirichenko on his 65th birthday and V. I. Sushchansky on his 60th birthday Abstract. We show that the class of groups generated by 3-state automata over a 2-letter alphabet has no more than 122 members. For each group in the class we provide some basic in- formation, such as short relators, a few initial values of the growth function, a few initial values of the sizes of the quotients by level stabilizers (congruence quotients), and hystogram of the spectrum of the adjacency operator of the Schreier graph of the action on level 9. In most cases we provide more information, such as whether the group is contracting, self-replicating, or (weakly) branch group, and exhibit elements of infinite order (we show that no group in the class is an infinite torsion group). A GAP package, written by Muntyan and Savchuk, was used to perform some necessary cal- culations. For some of the examples, we establish that they are (virtually) iterated monodromy groups of post-critically finite ra- tional functions, in which cases we describe the functions and the limit spaces. There are exactly 6 finite groups in the class (of or- der no greater than 16), two free abelian groups (of rank 1 and 2), and only one free nonabelian group (of rank 3). The other exam- ples in the class range from familiar (some virtually abelian groups, lamplighter group, Baumslag-Solitar groups BS(1,±3), and a free product C2 ∗ C2 ∗ C2) to enticing (Basilica group and a few other iterated monodromy groups). All authors were partially supported by at least one of the NSF grants DMS-308985, DMS-456185, DMS-600975, and DMS-605019 2000 Mathematics Subject Classification: 20E08. Key words and phrases: automata groups, self-similar groups, branch groups. 2 Classification of groups generated by automata 1. Introduction Automaton groups were formally introduced in the beginning of 1960’s [Glu61, Hoř63] but it took a while to realize their importance, utility, and, at the same time, complexity. Among the publications from the first decade of the study of automaton groups let us distin- guish [Zar64, Zar65] and the book [GP72]. The first substantial results came only in the 1970’s and in the be- ginning of the 1980’s when it was shown in [Ale72, Sus79, Gri80, GS83b] that automaton groups provide examples of finitely generated infinite torsion groups, thus making a contribution to one of the most fa- mous problems in algebra — the General Burnside Problem (more in- formation on all three versions of the Burnside problem can be found in [Adi79, Gol68, Gup89, Kos90, Zel91, GL02]). The methods used to study the properties of the examples from [Ale72, Sus79, Gri80] are very different. The methods used in [Ale72] are typical for the theory of fi- nite automata (in fact the provided proof was incorrect; the first correct proof appears in [Mer83] as a combination of the results from [Gri80] and [Mer83], as well as in the third edition of the book [KM82] and in [KAP85]). The exposition in [Sus79] is based on Kalujnin’s tableaux coming from his theory of iterated wreath products of cyclic groups of prime order p. The approach in [Gri80] is based on the ideas of self- similarity and contraction. These ideas are apparent both in the proof of the infiniteness and the torsion property of the group. The self-similarity is apparent from the fact that the set of all states of the automaton is used as a generating set for the group (now it is common to call such groups self-similar). The contraction property here means that the length of the elements contracts by a factor bounded away from 1 when one passes to sections. A principal tool introduced in the beginning of the 1980’s was the language of actions on rooted trees suggested by Gupta and Sidki in [GS83b], which helped tremendously in bringing geometric insight to the subject. A new indication of the importance of automaton groups came when it was shown that some of them provided the first examples of groups of intermediate growth [Gri83, Gri84, Gri85]. This not only answered the question of J.Milnor [Mil68] about existence of such groups, but also an- swered a number of other questions in and around group theory, including M. Day’s problem [Day57] on existence of amenable but not elementary amenable groups. Basically, even to this day, all known examples of groups of intermediate growth and non-elementary amenable groups are based on automaton groups. Investigations in the last two decades [Gri84, Gri85, GS83b, GS83a, I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Lys85, Neu86, Sid87a, Sid87b, Gri89, Roz93, Gri98, Gri99, Gri00, BG00a, BG00b, GŻ01, Nek05, GŠ06] show that many automaton groups pos- sess numerous interesting, and sometimes unusual, properties. This in- cludes just infiniteness (the groups constructed in [Gri84, Gri85] as well as in [GS83a] answer a question from [CM82] on new examples of infinite groups with finite quotients), finiteness of width, or more generally poly- nomial growth of the dimension of the successive quotients in the lower central series [BG00b] (answering a question of E. Zelmanov on classifi- cation of groups of finite width), branch properties [Gri84, Neu86, Gri00] (answering some questions of S. Pride and M. Edjvet [Pri80, EP84]), finiteness of the index of maximal subgroups and presence or absence of the congruence property [Per00, Per02] (related to topics in pro-finite groups), existence of groups with exponential but not uniformly expo- nential growth [Wil04b, Wil04a, Bar03, Nek07b] (providing an answer to a question of M. Gromov), subgroup separability and conjugacy separa- bility [GW00], further examples of amenable groups but not amenable (or even sub-exponentially amenable) groups [GŻ02a, BV05, GNŠ06a], amenability of groups generated by bounded automata [BKN], and so on. The word problem can be solved in contracting self-similar groups by using an extremely effective branch algorithm [Gri84, Sav03]. The con- jugacy problem can also be solved in many cases [WZ97, Roz98, Leo98, GW00] (in fact we do not know of an example of an automaton group with unsolvable conjugacy problem). In some instances, it is even known that the membership problem is solvable [GW03]. In addition to the formulation of many algebraic properties of groups generated by finite automata, a number of links and applications were discovered during the last decade. This includes asymptotic and spectral properties of the Cayley graphs and Schreier graphs associated to the action on the rooted tree with respect to the set of generators given by the set of states of the automaton. For instance, it is shown in [GŻ01] that the discrete Laplacian on the Cayley graph of the Lamplighter group Z ⋉ (Z/2Z)Z has pure point spectrum. This fact was used to answer a question of M. Atiyah on L2-Betti numbers of closed manifolds [GLSŻ00]. The methods developed in the study of the spectral properties of Schreier graphs of self-similar groups can be used to construct Laplacians on frac- tal sets and to study their spectral properties (see [GN07, NT08]. A new and fruitful direction, bringing further applications of self- similar groups, was established by the introduction of the notions of it- erated monodromy groups and limit spaces by V. Nekrashevych. The theory established a link between contracting self-similar groups and the geometry of Julia sets of expanding maps. An example of an application of self-similar groups to holomorphic dynamics is given by the solution 4 Classification of groups generated by automata (by L. Bartholdi and V. Nekrashevych in [BN06]) of the “twisted rab- bit” problem of J. Hubbard. The book [Nek05] provides a comprehensive introduction to this theory. In many situations automaton groups serve as renorm groups. For instance this happens in the study of classical fractals, in the study of the behavior of dynamical systems [Oli98], and in combinatorics — for example in Hanoi Towers games on k pegs, k ≥ 3, as observed by Z. Šunić (see [GŠ06]). There is interest of computer scientists and logicians in automaton groups, since they may be relevant in the solution of important complexity problems (see [RS] for ideas in this direction). Self-similar groups of intermediate growth are mentioned by Wolfram in [Wol02] as examples of “multiway systems” with complex behavior. Among the major problems in many areas of mathematics are the classification problems. If the objects are given combinatorially then it is naturally to try to classify them first by complexity and then within each complexity class. A natural complexity parameter in our situation is the pair (m,n) where m is the number of states of the automaton generating the group and n is the cardinality of the alphabet. There are 64 invertible 2-state automata acting on a 2-letter alphabet, but there are only six non-isomorphic (2, 2)-automaton groups, namely, the trivial group, Z/2Z, Z/2Z⊕Z/2Z, Z, the infinite dihedral group D∞, and the lamplighter group Z≀Z/2Z [GNS00, GŻ01] (more details are given in Theorem 7 below). A classification of semigroups generated by 2-state automata (not necessary invertible) over a 2-letter alphabet is provided by I. Reznikov and V. Sushchanskĭı [RS02a]. Some examples from this class, including an automaton generating a semigroup of intermediate growth, were studied in the subsequent papers [RS02c, RS02b, BRS06]. It is not known how many pairwise non-isomorphic groups exists for any class (m,n) when either m > 2 or n > 2. Unfortunately, the number of automata that has to be treated grows super-exponentially with either of the two arguments (there are mmn(n!)m invertible (m,n)-automata). Nevertheless, a reasonable task is to consider the problem of classifi- cation for small values of m and n and try to classify the (3, 2)-automaton groups and (2, 3)-automaton groups. Our research group (with some contribution by Y. Vorobets and M. Vorobets) has been working on the problem of classification of (3, 2)- automaton groups for the last four yeas and some of the obtained results are presented in this article. Our research goals moved in three main directions: 1. Search for new interesting groups and an attempt to use them to I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, solve known problems. An example of such a group is the Basilica group (see automaton [852]). It is the first example of an amenable group (shown in [BV05]) that is not sub-exponentially amenable group (shown in [GŻ02a]). 2. Recognition of already known groups as self-similar groups, and use of the self-similar structure in finding new results and applications for such groups. As examples we can mention the free group of rank 3 (see automaton [2240]), the free product of three copies of Z/2Z (see automa- ton [846]), Baumslag-Solitar groups BS(1,±3) (see automata [870] and [2294]), the Klein bottle group (see automaton [2212]), and the group of orientation preserving automorphisms of the 2-dimensional integer lattice (see automaton [2229]). 3. Understanding of typical phenomena that occur for various classes of automaton groups, formulation and proofs of reasonable conjectures about the structure of self-similar groups. The results on the class of groups generated by (3, 2)-automata proven in this article are the following. Theorem 1. There are at most 122 non-isomorphic groups generated by (3, 2)-automata. The numbers in brackets in the next two theorems are references to the numbers of the corresponding automata (more on this encoding will be said later). Here and thereafter, Cn denotes the cyclic group of order n. Theorem 2. There are 6 finite groups in the class: the trivial group {1} [1], C2 [1090], C2 × C2 [730], D4 [847], C2 × C2 × C2 [802] and D4 × C2 [748]. Theorem 3. There are 6 abelian groups in the class: the trivial group {1} [1], C2 [1090], C2 × C2 [730], C2 × C2 × C2 [802], Z [731] and Z 2 [771]. Theorem 4. The only nonabelian free group in the class is the free group of rank 3 generated by the Aleshin-Vorobets-Vorobets automaton [2240]. Theorem 5. There are no infinite torsion groups in the class. The short list of general results does not give full justice to the work that has been done. Namely, in most individual cases we have provided detailed information for the group in question. More work and, likely, some new invariants are required to further distinguish the 122 groups that are listed in this paper as potentially 6 Classification of groups generated by automata non-isomorphic. In some cases one could try to use the rigidity of actions on rooted trees (see [LN02]), since in many cases it is easier to distinguish actions than groups. In the contracting case one could use, for instance, the geometry of the Schreier graphs and limit spaces to distinguish the actions. Next natural step would be to consider the case of (2, 3)-automaton groups or 2-generated self-similar groups of binary tree automorphisms defined by recursions in which every section is either trivial, a generator, or an inverse of a generator. The cases (4, 2) and (5, 2) also seem to be attractive, as there are many remarkable groups in these classes. Another possible direction is to study more carefully only certain classes of automata (such as the classical linear automata, bounded and polynomially growing automata in the sense of Sidki [Sid00], etc.) and the properties of the corresponding automaton groups. Many computations used in our work were performed by the package AutomGrp for GAP system, developed by Y. Muntyan and D. Savchuk [MS08]. The package is not specific to (3, 2)-automaton groups (in fact, many functions are implemented also for groups of tree automorphisms that are not necessarily generated by automata). 2. Regular rooted trees, automorphisms, and self- similarity Let X be an alphabet on d (d ≥ 2) letters. Most often we set X = {0, 1, . . . , d − 1}. The set of finite words over X, denote by X∗, has the structure of a regular rooted d-ary tree, which we also denote by X∗. The empty word ∅ is the root of the tree and every vertex v has d children, namely the words vx, for x in X. The words of length n constitute level n in the tree. The group of tree automorphisms of X∗ is denoted by Aut(X∗). Tree automorphisms are precisely the permutations of the vertices that fix the root and preserve the levels of the tree. Every automorphism f of X∗ can be decomposed as f = αf (f0, . . . , fd−1) (1) where fx, for x in X, are automorphisms of X∗ and αf is a permutation of the set X. The permutation αf is called the root permutation of f and the automorphisms fx (denoted also by f \|x), x in X, are called sections of f . The permutation αf describes the action of f on the first letter of every word, while the automorphism fx, for x in X, describes the action of f on the tail of the words in the subtree xX∗, consisting of the words I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, in X∗ that start with x. Thus the equality (1) describes the action of f through decomposition into two steps. In the first step the d-tuple (f0, . . . , fd−1) acts on the d subtrees hanging below the root, and then the permutation αf , permutes these d subtrees. Thus we have f(xw) = αf (x)fx(w), (2) for x in X and w in X∗. Second level sections of f are defined as the sections of the sections of f , i.e., fxy = (fx)y, for x, y ∈ X, and more generally, for a word u in X∗ and a letter x in X the section of f at ux is defined as fux = (fu)x, while the section of f at the root is f itself. The group Aut(X∗) decomposes algebraically as Aut(X∗) = Sym(X) ⋉ Aut(X∗)X = Sym(X) ≀ Aut(X∗), (3) where ≀ is the permutational wreath product in which the ac- tive group Sym(X) permutes the coordinates of Aut(X∗)X = (Aut(X∗), . . . ,Aut(X∗)). For arbitrary automorphisms f and g in Aut(X∗) we have αf (f0, . . . , fd−1)αg(g0, . . . , gd−1) = αfαg(fg(0)g0, . . . , fg(d−1)gd−1). For future use we note the following formula regarding the sections of a composition of tree automorphisms. For tree automorphisms f and g and a vertex u in X∗, (fg)u = fg(u)gu. (4) The group of tree automorphisms Aut(X∗) is a pro-finite group. Namely, Aut(X∗) has the structure of an infinitely iterated wreath prod- uct Aut(X∗) = Sym(X) ≀ (Sym(X) ≀ (Sym(X) ≀ . . . )) of the finite group Sym(X∗) (this follows from (3)). This product is the inverse limit of the sequence of finitely iterated wreath products of the form Sym(X) ≀ (Sym(X) ≀ (Sym(X) ≀ · · · ≀ Sym(X))). Every subgroup of Aut(X∗) is residually finite. A canonical sequence of normal subgroups of finite index intersecting trivially is the sequence of level stabilizers. The n-th level stabilizer of a group G of tree automorphisms is the subgroup StabG(n) of Aut(X∗) that consists of all tree automorphisms in G that fix the vertices in the tree X∗ up to and including level n. The boundary of the tree X∗ is the set Xω of right infinite words over X (infinite geodesic rays in X∗ connecting the root to “infinity”). The boundary has a natural structure of a metric space in which two infinite words are close if they agree on long finite prefixes. More precisely, for 8 Classification of groups generated by automata two distinct rays ξ and ζ, define the distance to be d(ξ, ζ) = 1/2\|ξ∧ζ\|, where \|ξ ∧ ζ\| denotes the length of the longest common prefix ξ ∧ ζ of ξ and ζ. The induced topology on Xω is the Tychonoff product topology (with X discrete), and Xω is a Cantor set. The group of isometries Isom(Xω) and the group of tree automorphisms Aut(X∗) are canonically isomorphic. Namely, the action of the automorphism group Aut(X∗) can be extended to an isometric action on Xω, simply by declaring that (1) and (2) are valid for right infinite words. We now turn to the concept of self-similarity. The tree X∗ is a highly self-similar object (the subtree uX∗ consisting of words with prefix u is canonically isomorphic to the whole tree) and we are interested in groups of tree automorphisms in which this self-similarity structure is reflected. Definition 1. A group G of tree automorphisms is self-similar if, every section of every automorphism in G is an element of G. Equivalently, self-similarity can be expressed as follows. A group G of tree automorphisms is self-similar if, for every g in G and a letter x in X, there exists a letter y in X and an element h in G such that g(xw) = yh(w), for all words w over X. Self-replicating groups constitute a special class of self-similar groups. Examples from this class are very common in applications. A self-similar group G is self-replicating if, for every vertex u in X∗, the homomorphism ϕu : StabG(u) → G from the stabilizer of the vertex u in G to G, given by ϕ(g) = gu, is surjective. At the end of the section, let us mention the class of branch groups. Branch groups were introduced [Gri00] where it is shown that they con- stitute one of the three classes of just-infinite groups (infinite groups with no proper, infinite, homomorphic images). If a class of groups C is closed under homomorphic images and if it contains infinite, finitely gen- erated examples then it contains just-infinite examples (this is because every infinite, finitely generated group has a just-infinite image). Such examples are minimal infinite examples in C. We note that, for exam- ple, the group of intermediate growth constructed in [Gri80] is a branch automaton group that is a just-infinite 2-group. i.e., it is an infinite, finitely generated, torsion group that has no proper infinite quotients. The Hanoi Towers group [GŠ07] is a branch group that is not just infi- nite [GNŠ06b]. The iterated monodromy group IMG(z2 + i) [GSŠ07] is a branch groups, while B = IMG(z2−1) is not a branch group, but only weakly branch. More generally, it is shown in [BN07] that the iterated I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, monodromy groups of post-critically finite quadratic maps are branch groups in the pre-periodic case and weakly branch groups in the periodic case (the case refers to the type of post-critical behavior). We now define regular (weakly) branch groups. A level transitive group G ≤ Aut(X∗) of k-ary tree automorphisms is a regular branch group over K if K is a normal subgroup of finite index in G such that K×· · ·×K is geometrically contained in K. By definition, the subgroup K has the property that K × · · · × K is geometrically contained in K, denoted by K × · · · ×K � K, if K × · · · ×K ≤ ψ(K ∩ StabG(1)) where ψ is the homomorphism ψ : StabG(1) → Aut(X∗)×· · ·×Aut(X∗) given by ψ(g) = (g0, g1, . . . , gk−1). If instead of asking for K to have finite index in G we only require that K is nontrivial, we say that G is regular weakly branch group over K. Note that if G is level transitive and K is normal in G, in order to show that G is regular (weakly) branch group over K, it is sufficient to show that K × 1 × · · · × 1 � K (i.e. K × 1 × · · · × 1 ≤ ψ(K ∩ StabG(1))). More on the class of branch group can be found in [Gri00] and [BGŠ03]. 3. Automaton groups The full group of tree automorphisms Aut(X∗) is self-similar, since the section of every tree automorphism is just another tree automorphism. However, this group is rather large (uncountable). For various reasons, one may be interested in ways to define (construct) finitely generated self- similar groups. Automaton groups constitute a special class of finitely generated self-similar groups. We provide two ways of thinking about automaton groups. One is through finite wreath recursions and the other through finite automata. Every finite system of recursive relations of the form        s(1) = α1 ( s (1) 0 , s (1) 1 , . . . , s (1) d−1 ) , . . . s(k) = αk ( s (k) 0 , s (k) 1 , . . . , s (k) d−1 ) , (5) where each symbol s (i) j , i = 1, . . . , k, j = 0, . . . , d−1, is a symbol in the set of symbols {s(1), . . . , s(k)} and α1, . . . , αk are permutations in Sym(X), has a unique solution in Aut(X∗) (in the sense that the above recur- sive relations represent the decompositions of the tree automorphisms 10 Classification of groups generated by automata s(1), . . . , s(k)). Thus, the action of the automorphism defined by the sym- bol s(i) is given recursively by s(i)(xw) = αi(x)s (i) x (w). The group G generated by the automorphisms s(1), . . . , s(k) is a finitely generated self-similar group of automorphisms of X∗. This fol- lows since sections of products are products of sections (see (4)) and all sections of the generators of G are generators of G. When a self-similar group is defined by a system of the form (5), we say that it is defined by a wreath recursion. We switch now the point of view from wreath recursions to invertible automata. Definition 2. A finite automaton A is a 4-tuple A = (S,X, π, τ) where S is a finite set of states, X is a finite alphabet of cardinality d ≥ 2, π : S×X → X is a map, called output map, and τ : S×X → S is a map, called transition map. If in addition, for each state s in S, the restriction πs : X → X given by πs(x) = π(s, x) is a permutation in Sym(X), the automaton A is invertible. In fact, we will be only concerned with finite invertible automata and, in the rest of the text, we will use the word automaton for such automata. Each state s of the automaton A defines a tree automorphism of X∗, which we also denote by s. By definition, the root permutation of the automorphism s (defined by the state s) is the permutation πs and the section of s at x is τ(s, x). Therefore s(xw) = πs(x)τ(s, x)(w) (6) for every state s in S, letter x in X and word w over X. Definition 3. Given an automaton A = (S,X, π, τ), the group of tree automorphisms generated by the states of A is denoted by G(A) and called the automaton group defined by A. The generating set S of the automaton group G(A) generated by the automaton A = (S,X, π, τ) is called the standard generating set of G(A) and plays a distinguished role. Directed graphs provide convenient representation of automata. The vertices of the graph, called Moore diagram of the automaton A = (S,X, π, τ), are the states in S. Each state s is labeled by the root permutation αs = πs and, for each pair (s, x) ∈ S ×X, an edge labeled by x connects s to sx = τ(s, x). Several examples are presented in Fig- ure 1. The states of the 5-state automaton in the left half of the figure generate the group G of intermediate growth mentioned in the introduc- tion (σ denotes the permutation exchanging 0 and 1, and 1 denotes the trivial vertex permutation). The top of the three 2-state automata on the I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych,PSfrag repla ements 0000 0 00 01 111 11 11 1 11 1 1 1 1 110; 10; 1 0; 1aa a a bb b d y �� Figure 1: An automaton generating G, the binary adding machine, and two Lamplighter automata right in Figure 1 is the so called binary adding machine, which generates the infinite cyclic group Z. The other two automata both generate the Lamplighter group L2 = Z ≀ Z/2Z = Z ⋉ ( ⊕ Z/2Z)Z (see [GNS00]). The corresponding wreath recursions for the adding machine and for the two automata generating the Lamplighter group are given by a = σ(1, a) a = σ(b, a) a = (b, a) 1 = (1, 1) b = (b, a), b = σ(a, b) (7) respectively. The class of polynomially growing automata was introduced by Sidki in [Sid00]. Sidki proved in [Sid04] that no group generated by such an automaton contains free subgroups of rank 2. As we already indicated in the introduction, for the subclass of so called bounded automata the corresponding groups are amenable [BKN]. Recall that an automaton A is called bounded if, for every state s of A, the function fs(n) counting the number of active sections of s at level n is bounded (a state is active if its vertex permutation is nontrivial). There are other classes of automata (and corresponding automaton groups) that deserve special attention. We end the section by mentioning several such classes. The class of linear automata consists of automata in which both the set of states S and the alphabet X have a structure of a vector space (over a finite field) and both the output and the transition function are linear maps (see [GP72] and [Eil76]). The class of bi-invertible automata consists of automata in which both the automaton and its dual are invertible. Some of the automata in our classification are bi-invertible, most notably the Aleshin-Vorobets- Vorobets automaton [2240] generating the free group F3 of rank 3 and 12 Classification of groups generated by automata the Bellaterra automaton [846] generating the free product C2 ∗C2 ∗C2. In fact, both of these have even stronger property of being fully invertible. Namely, not only the automaton and its dual are invertible, but also the dual of the inverse automaton is invertible. Another important class is the class of automata satisfying the open set condition. Every automaton in this class contains a trivial state (a state defining the trivial tree automorphism) and this state can be reached from any other state. One may also study automata that are strongly connected (i.e. au- tomata for which the corresponding Moore diagrams are strongly con- nected as directed graphs), automata in which no path contains more than one active state (such as the automaton defining G in Figure 1), and so on. 4. Schreier graphs Let G be a group generated by a finite set S and let G act on a set Y . We denote by Γ = Γ(G,S, Y ) the Schreier graph of the action of G on Y . The vertices of Γ are the elements of Y . For every pair (s, y) in S × Y an edge labeled by s connects y to s(y). An orbital Schreier graph of the action is the Schreier graph Γ(G,S, y) of the action of G on the G-orbit of y, for some y in Y . Let G be a group of tree automorphisms of X∗ generated by a fi- nite set S. The levels Xn, n ≥ 0, are invariant under the action of G and we can consider the sequence of finite Schreier graphs Γn(G,S) = Γ(G,S,Xn), n ≥ 0. Let ξ = x1x2x3 . . . ∈ Xω be an infinite ray. Then the pointed Schreier graphs (Γn(G,S), x1x2 . . . xn) converge in the local topology (see [Gri84] or [GŻ99]) to the pointed orbital Schreier graph (Γ(G,S, ξ), ξ). Schreier graphs may be sometimes used to compute the spectrum of some operators related to the group. For a group of tree automorphisms G generated by a finite symmetric set S there is a natural unitary rep- resentation in the space of bounded linear operators H = B(L2(X ω)), given by πg(f)(x) = f(g−1x) (the measure on the boundary Xω is just the product measure associated to the uniform measure on X). Consider the spectrum of the operator M = 1 \|S\| ∑ s∈S πs corresponding to this unitary representation. The spectrum of M for a self-similar group G is approximated by the spectra of the finite di- mensional operators induced by the action of G on the levels of the tree I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, (see [BG00a]. Denote by Hn the subspace of H = B(L2(X ω)) spanned by the characteristic functions fv, v ∈ Xn, of the cylindrical sets corre- sponding to the \|X\|n vertices on level n. The subspace Hn is invariant under the action of G and Hn ⊂ Hn+1. Denote by π (n) g the restriction of πg on Hn. Then, for n ≥ 0, the operator Mn = 1 \|S\| ∑ s∈S π(n) s is finite dimensional. Moreover, sp(M) = ⋃ n≥0 sp(Mn), i.e., the spectra of the operators Mn converge to the spectrum of M . The table of information provided in Section 8 includes, in each case, the histogram of the spectrum of the operator M9. If P is the stabilizer of a point on the boundary Xω, then one can consider the quasi-regular representation ρG/P of G in ℓ2(G/P ). Theorem 6 ([BG00a]). If G is amenable or the Schreier graph G/P (the Schreier graph of the action of G on the cosets of P ) is amenable then the spectrum of M and the spectrum of the quasi-regular representation ρG/P coincide. In case the parabolic subgroup P is “small”, the last result may be used to compute the spectrum of the Markov operator on the Cayley graph of the group. This approach was successfully used, for instance, to compute the spectrum of the Lamplighter group in [GŻ01] (see also [KSS06]). 5. Contracting groups, limit spaces, and iterated mon- odromy groups Definition 4. A group G generated by an automaton over alphabet X is contracting if there exists a finite subset N ⊂ G such that for every g ∈ G there exists n (generally depending on g) such that section gv belongs to N for all words v ∈ X∗ of length at least n. The smallest set N with this property is called the nucleus of the group G. The above definition makes sense for arbitrary self-similar groups — not necessarily automaton groups and, moreover, not necessarily finitely generated groups. In the case of an automaton group the contracting property may be equivalently stated as follows. An automaton group G = G(A) is contracting if there exist constants κ, C, and N , with 14 Classification of groups generated by automata 0 ≤ κ < 1, such that \|gv\| ≤ κ\|g\| + C, for all vertices v of length at least N and g ∈ G (the length is measured with respect to the standard generating set S consisting of the states of A). The contraction property is a key ingredient in many inductive arguments and algorithms involving the decomposition g = αg(g0, . . . , gd−1). Indeed, the contraction property implies that, for all sufficiently long elements g, all sections of g at vertices on level at least N are strictly shorter than g. Contracting groups have rich geometric structure. Each contracting group is the iterated monodromy group of its limit dynamical system. This system is an (orbispace) self-covering of the limit space of the group. The limit space is a limit of the graphs of the action of G on the levels Xn of the tree X∗ and is defined in the following way. Definition 5. Let G be a contracting group over X. Denote by X−ω the space of all left-infinite sequences . . . x2x1 of elements ofX with the direct product (Tykhonoff) topology. We say that two sequences . . . x2x1 and . . . y2y1 are asymptotically equivalent if there exists a sequences gk ∈ G, assuming a finite set of values, and such that gk(xk . . . x1) = yk . . . y1 for all k ≥ 1. The quotient of the space X−ω by this equivalence relation is called the limit space of G. The following proposition, proved in [Nek05] (Proposition 3.6.4) is a convenient way to compute the asymptotic equivalence. Proposition 1. Let a contracting group G be generated by a finite au- tomaton A. Then the asymptotic equivalence is the equivalence relation generated by the set of pairs (. . . x2x1, . . . y2y1) for which there exists a sequence gk of states of A such that gk(xk) = yk and gk\|xk = gk−1. The limit dynamical system is the map induced by the shift . . . x2x1 7→ . . . x3x2. The limit space is a compact metrizable topological space of fi- nite topological dimension (see [Nek05], Theorem 3.6.3). If the group is self-replicating, then the limit space is locally connected and path con- nected. The main tool of finding the limit space of a contracting group is realization of the group as the iterated monodromy group of an expand- ing partial orbispace self-covering. An exposition of the theory of such self-coverings is given in [Nek05]. In particular, if G is the iterated mon- odromy group of a post-critically finite complex rational function, then the limit space of G is homeomorphic to the Julia set of the function (see Theorems 5.5.3 and 6.4.4 of [Nek05]). I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, The limit space does not change when we pass from X to Xn in the natural way (we will change then the limit dynamical system to its nth iterate). It also does not change if we post-conjugate the wreath recursion by an element of the wreath product Symm(X)⋉GX , i.e., conjugate the group G by an element of the form γ = π(g0γ, g1γ), where π ∈ Symm(X) and g0, g1 ∈ G. The limit space can be also visualized using its subdivision into tiles. This method is especially effective, when the group is generated by bounded automata. Definition 6. Let G be a contracting group. A tile TG of G is the quotient of the space X−ω by the equivalence relation, which identifies two sequences . . . x2x1 and . . . y2y1 if there exists a sequence gk ∈ G assuming a finite number of values and such that gk(xk . . . x1) = yk . . . y1, gk\|xk...x1 = 1 for all k. Again, an analog of Proposition 1 is true: the equivalence relation from Definition 6 is generated by the identifications . . . x2x1 = . . . y2y1 of sequences for which there exists a sequence gk, k = 0, 1, 2, . . . of elements of the nucleus such that gk(xk) = yk, gk\|xk = gk−1 and g0 = 1. Suppose that G satisfies the open set condition, i.e., the trivial state can be reached from any other state of the generating automaton. Then the boundary of the tile TG is the image in TG of the set of sequences . . . x2x1 such that there exists a sequence gk ∈ G assuming a finite number of values and such that gk\|xk...x1 6= 1. If G is generated by a finite symmetric set S, then it is sufficient to look for the sequence gk inside S. The limit space of G is obtained from the tile by some identifications of the points of the boundary. If the group G is generated by bounded automata, then its boundary consists of a finite number of points and it is not hard to identify them (i.e., to identify the sequences encoding them). For v ∈ Xn denote by TGv the image of the cylindrical set X−ωv in TG. It is easy to see that the map . . . x2x1 7→ . . . x2x1v induces a homeomorphism of TG with TGv and that TG = ⋃ v∈Xn TGv. It is proved in [Nek05] that two pieces TGv1 and TGv2 intersect if and only if g(v1) = v2 for an element g of the nucleus of G and that they intersect only along images of the boundary of TG. 16 Classification of groups generated by automata This suggests the following procedure of visualizing the limit space in the case of bounded automata. Identify the points of the boundary of the tile. We get a finite list B of points, represented by a finite list W of infinite sequences (some points may be represented by several sequences). Draw the tile as a graph with \|B\| “boundary points” (vertices) and identify the boundary points with the points of B labeled by sequences W . Take now \|X\| copies of this tile, corresponding to different letters ofX. Append the corresponding letters x ∈ X to the ends of the labels w ∈ W of the boundary points of each of the copy of the tile. Some of the obtained labels will be related by the equivalence relation of Definition 6, i.e., represent the same points of the tile TG. Glue the corresponding points together. Some of the obtained labels will belong to W . These points will be the new boundary points. In this way we get a new graph with labeled boundary points. Repeat now the procedure several times, rescaling the graph in such a way that the original first order graphs become small. We will get in this way a graph resembling the tile TG (see Chapter V in [Bon07] for more details). Making the necessary identifications of its boundary we get an approximation of the limit space of G. More details on this inductive approximation procedure can be found in [Nek05] Section 3.10. The limit space of a finitely generated contracting self-similar group G can also be viewed as a hyperbolic boundary in the following way. For a given finite generating system S of G define the self-similarity graph Σ(G,S) as the graph with set of vertices X∗ in which two vertices v1, v2 ∈ X∗ are connected by an edge if and only if either vi = xvj , for some x ∈ X (vertical edges), or s(vi) = vj for some s ∈ S (horizontal edges). In case of a contracting group, the self-similarity graph Σ(G,S) is Gromov-hyperbolic and its hyperbolic boundary is homeomorphic to the limit space JG. The iterated monodromy group (IMG) construction is dual to the limit space construction. It may be defined for partial self-coverings of orbispaces, but we will only provide the definition in case of topological spaces, since we do not need the more general construction in this text (all iterated monodromy groups that appear later are related to partial self-coverings of the Riemann sphere). Let M be a path connected and locally path connected topological space and let M1 be an open path connected subset of M. Let f : M1 → M be a d-fold covering. Denote by fn the n-fold iteration of the map f . Then fn : Mn → M, where Mn = f−n(M), is a dn-fold covering. Fix a base point t ∈ M and let Tt be the disjoint union of the sets f−n(t), n ≥ 0 (formally speaking, these sets may not be disjoint in M). The set of pre-images Tt has a natural structure of a rooted d-ary tree. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, The base point t is the root, the vertices in f−n constitute level n and every vertex z in f−n(t) is connected by an edge to f(z) in f−n+1(t), for n ≥ 1. The fundamental group π1(M, t) acts naturally, through the mon- odromy action, on every level f−n(t) and, in fact, acts by automorphisms on Tt. Definition 7. The iterated monodromy group IMG(f) of the covering f is the quotient of the fundamental group π1(M, t) by the kernel of its action on the tree of pre-images Tt. 6. Classification guide Every 3-state automaton A with set of states S = {0,1,2} acting on the 2-letter alphabet X = {0, 1} is assigned a unique number as follows. Given the wreath recursion    0 = σa11(a12, a13), 1 = σa21(a22, a23), 2 = σa31(a32, a33), defining the automaton A, where aij ∈ {0,1,2} for j 6= 1 and ai1 ∈ {0, 1}, i = 1, 2, 3, assign the number Number(A) = a12 + 3a13 + 9a22 + 27a23 + 81a32+ 243a33 + 729(a11 + 2a21 + 4a31) + 1 to A. With this agreement every (3, 2)-automaton is assigned a unique number in the range from 1 to 5832. The numbering of the automata is induced by the lexicographic ordering of all automata in the class. Each of the automata numbered 1 through 729 generates the trivial group, since all vertex permutations are trivial in this case. Each of the automata numbered 5104 through 5832 generates the cyclic group C2 of order 2, since both states represent the automorphism that acts by changing all letters in every word over X. Therefore the nontrivial part of the classi- fication is concerned with the automata numbered by 730 through 5103. Denote by An the automaton numbered by n and by Gn the corre- sponding group of tree automorphisms. Sometimes we may use just the number to refer to the corresponding automaton or group. The following three operations on automata do not change the iso- morphism class of the group generated by the corresponding automaton (and do not change the action on the tree in essential way): (i) passing to inverses of all generators, 18 Classification of groups generated by automata (ii) permuting the states of the automaton, (iii) permuting the alphabet letters. Definition 8. Two automata A and B that can be obtained from one another by using a composition of the operations (i)–(iii), are called symmetric. For instance, the two automata in the lower right part of Figure 1 are symmetric. The wreath recursion for the automaton obtained by permuting both the names of the states and the alphabet letters of the first of these two automata is a = (b, a) b = σ(b, a) and this wreath recursion describes exactly the inverses of the tree auto- morphism defining the second of the two automata. Additional identifications can be made after automata minimization is applied. Definition 9. If the minimization of an automaton A is symmetric to the minimization of an automaton B, we say that the automata A and B are minimally symmetric and write A ∼ B. There are 194 classes of (3, 2)-automata that are pairwise not mini- mally symmetric. Of these, 10 are minimally symmetric to automata with fewer than 3 states and, as such, are subject of Theorem 7 ([GNS00], see below). At present, it is known that there are no more than 122 non- isomorphic (3, 2)-automaton groups. Some information on these groups is given in Section 8. The proofs of some particular properties of the 194 classes of non- equivalent automata (and in particular, all known isomorphisms) can be found in Section 9. The few general results that hold in the whole class were already mentioned in the introduction. The table in Section 7 may be used to determine the equivalence and the group isomorphism class for each automaton. Every class is numbered by the smallest number of an automaton in the class. For instance, an entry such as x ∼ y ∼= z means that the automata with the smallest number in the equivalence and the (known) isomorphism class of x are y and z, respectively. While the equivalence classes are easy to determine the isomorphism class is not. Therefore, there may still be some additional isomorphisms between some of the classes (which would I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, eventually cause changes in the z numbers and consolidation of some of the current isomorphism classes). If one is interested in some particular (3, 2)-automaton A, we recom- mend the following procedure: • Use the table in Section 7 to find numbers for the representatives of the equivalence and the isomorphism class of A. Minimizing the automaton and finding the symmetry is a straightforward task, which is not presented here. • Use Section 8 to find information on the group generated by A (more precisely, the isomorphic group generated by the chosen rep- resentative in the class). • Use Section 9 to find the proof of the isomorphism and some known properties. 20 Classification of groups generated by automata 7. Table of equivalence classes (and known isomorphisms) For explanation of the entries see Section 6. 1 through 729 ∼ 1 ≃ 1, 730 ∼ 730 ∼= 730 731 ∼ 731 ∼= 731 732 ∼ 731 ∼= 731 733 ∼ 731 ∼= 731 734 ∼ 734 ∼= 730 735 ∼ 734 ∼= 730 736 ∼ 731 ∼= 731 737 ∼ 734 ∼= 730 738 ∼ 734 ∼= 730 739 ∼ 739 ∼= 739 740 ∼ 740 ∼= 740 741 ∼ 741 ∼= 741 742 ∼ 740 ∼= 740 743 ∼ 743 ∼= 739 744 ∼ 744 ∼= 744 745 ∼ 741 ∼= 741 746 ∼ 744 ∼= 744 747 ∼ 747 ∼= 739 748 ∼ 748 ∼= 748 749 ∼ 749 ∼= 749 750 ∼ 750 ∼= 750 751 ∼ 749 ∼= 749 752 ∼ 752 ∼= 752 753 ∼ 753 ∼= 753 754 ∼ 750 ∼= 750 755 ∼ 753 ∼= 753 756 ∼ 756 ∼= 748 757 ∼ 739 ∼= 739 758 ∼ 740 ∼= 740 759 ∼ 741 ∼= 741 760 ∼ 740 ∼= 740 761 ∼ 743 ∼= 739 762 ∼ 744 ∼= 744 763 ∼ 741 ∼= 741 764 ∼ 744 ∼= 744 765 ∼ 747 ∼= 739 766 ∼ 766 ∼= 730 767 ∼ 767 ∼= 731 768 ∼ 768 ∼= 731 769 ∼ 767 ∼= 731 770 ∼ 770 ∼= 730 771 ∼ 771 ∼= 771 772 ∼ 768 ∼= 731 773 ∼ 771 ∼= 771 774 ∼ 774 ∼= 730 775 ∼ 775 ∼= 775 776 ∼ 776 ∼= 776 777 ∼ 777 ∼= 777 778 ∼ 776 ∼= 776 779 ∼ 779 ∼= 779 780 ∼ 780 ∼= 780 781 ∼ 777 ∼= 777 782 ∼ 780 ∼= 780 783 ∼ 783 ∼= 775 784 ∼ 748 ∼= 748 785 ∼ 749 ∼= 749 786 ∼ 750 ∼= 750 787 ∼ 749 ∼= 749 788 ∼ 752 ∼= 752 789 ∼ 753 ∼= 753 790 ∼ 750 ∼= 750 791 ∼ 753 ∼= 753 792 ∼ 756 ∼= 748 793 ∼ 775 ∼= 775 794 ∼ 776 ∼= 776 795 ∼ 777 ∼= 777 796 ∼ 776 ∼= 776 797 ∼ 779 ∼= 779 798 ∼ 780 ∼= 780 799 ∼ 777 ∼= 777 800 ∼ 780 ∼= 780 801 ∼ 783 ∼= 775 802 ∼ 802 ∼= 802 803 ∼ 803 ∼= 771 804 ∼ 804 ∼= 731 805 ∼ 803 ∼= 771 806 ∼ 806 ∼= 802 807 ∼ 807 ∼= 771 808 ∼ 804 ∼= 731 809 ∼ 807 ∼= 771 810 ∼ 810 ∼= 802 811 ∼ 748 ∼= 748 812 ∼ 750 ∼= 750 813 ∼ 749 ∼= 749 814 ∼ 750 ∼= 750 815 ∼ 756 ∼= 748 816 ∼ 753 ∼= 753 817 ∼ 749 ∼= 749 818 ∼ 753 ∼= 753 819 ∼ 752 ∼= 752 820 ∼ 820 ∼= 820 821 ∼ 821 ∼= 821 822 ∼ 821 ∼= 821 823 ∼ 821 ∼= 821 824 ∼ 824 ∼= 820 825 ∼ 824 ∼= 820 826 ∼ 821 ∼= 821 827 ∼ 824 ∼= 820 828 ∼ 824 ∼= 820 829 ∼ 820 ∼= 820 830 ∼ 821 ∼= 821 831 ∼ 821 ∼= 821 832 ∼ 821 ∼= 821 833 ∼ 824 ∼= 820 834 ∼ 824 ∼= 820 835 ∼ 821 ∼= 821 836 ∼ 824 ∼= 820 837 ∼ 824 ∼= 820 838 ∼ 838 ∼= 838 839 ∼ 839 ∼= 821 840 ∼ 840 ∼= 840 841 ∼ 839 ∼= 821 842 ∼ 842 ∼= 838 843 ∼ 843 ∼= 843 844 ∼ 840 ∼= 840 845 ∼ 843 ∼= 843 846 ∼ 846 ∼= 846 847 ∼ 847 ∼= 847 848 ∼ 848 ∼= 750 849 ∼ 849 ∼= 849 850 ∼ 848 ∼= 750 851 ∼ 851 ∼= 847 852 ∼ 852 ∼= 852 853 ∼ 849 ∼= 849 854 ∼ 852 ∼= 852 855 ∼ 855 ∼= 847 856 ∼ 856 ∼= 856 857 ∼ 857 ∼= 857 858 ∼ 858 ∼= 858 859 ∼ 857 ∼= 857 860 ∼ 860 ∼= 860 861 ∼ 861 ∼= 861 862 ∼ 858 ∼= 858 863 ∼ 861 ∼= 861 864 ∼ 864 ∼= 864 865 ∼ 865 ∼= 820 866 ∼ 866 ∼= 866 867 ∼ 866 ∼= 866 868 ∼ 866 ∼= 866 869 ∼ 869 ∼= 869 870 ∼ 870 ∼= 870 871 ∼ 866 ∼= 866 872 ∼ 870 ∼= 870 873 ∼ 869 ∼= 869 874 ∼ 874 ∼= 874 875 ∼ 875 ∼= 875 876 ∼ 876 ∼= 876 877 ∼ 875 ∼= 875 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 878 ∼ 878 ∼= 878 879 ∼ 879 ∼= 879 880 ∼ 876 ∼= 876 881 ∼ 879 ∼= 879 882 ∼ 882 ∼= 882 883 ∼ 883 ∼= 883 884 ∼ 884 ∼= 884 885 ∼ 885 ∼= 885 886 ∼ 884 ∼= 884 887 ∼ 887 ∼= 887 888 ∼ 888 ∼= 888 889 ∼ 885 ∼= 885 890 ∼ 888 ∼= 888 891 ∼ 891 ∼= 891 892 ∼ 739 ∼= 739 893 ∼ 741 ∼= 741 894 ∼ 740 ∼= 740 895 ∼ 741 ∼= 741 896 ∼ 747 ∼= 739 897 ∼ 744 ∼= 744 898 ∼ 740 ∼= 740 899 ∼ 744 ∼= 744 900 ∼ 743 ∼= 739 901 ∼ 820 ∼= 820 902 ∼ 821 ∼= 821 903 ∼ 821 ∼= 821 904 ∼ 821 ∼= 821 905 ∼ 824 ∼= 820 906 ∼ 824 ∼= 820 907 ∼ 821 ∼= 821 908 ∼ 824 ∼= 820 909 ∼ 824 ∼= 820 910 ∼ 820 ∼= 820 911 ∼ 821 ∼= 821 912 ∼ 821 ∼= 821 913 ∼ 821 ∼= 821 914 ∼ 824 ∼= 820 915 ∼ 824 ∼= 820 916 ∼ 821 ∼= 821 917 ∼ 824 ∼= 820 918 ∼ 824 ∼= 820 919 ∼ 919 ∼= 820 920 ∼ 920 ∼= 920 921 ∼ 920 ∼= 920 922 ∼ 920 ∼= 920 923 ∼ 923 ∼= 923 924 ∼ 924 ∼= 870 925 ∼ 920 ∼= 920 926 ∼ 924 ∼= 870 927 ∼ 923 ∼= 923 928 ∼ 928 ∼= 820 929 ∼ 929 ∼= 929 930 ∼ 930 ∼= 821 931 ∼ 929 ∼= 929 932 ∼ 932 ∼= 820 933 ∼ 933 ∼= 849 934 ∼ 930 ∼= 821 935 ∼ 933 ∼= 849 936 ∼ 936 ∼= 820 937 ∼ 937 ∼= 937 938 ∼ 938 ∼= 938 939 ∼ 939 ∼= 939 940 ∼ 938 ∼= 938 941 ∼ 941 ∼= 941 942 ∼ 942 ∼= 942 943 ∼ 939 ∼= 939 944 ∼ 942 ∼= 942 945 ∼ 945 ∼= 941 946 ∼ 838 ∼= 838 947 ∼ 840 ∼= 840 948 ∼ 839 ∼= 821 949 ∼ 840 ∼= 840 950 ∼ 846 ∼= 846 951 ∼ 843 ∼= 843 952 ∼ 839 ∼= 821 953 ∼ 843 ∼= 843 954 ∼ 842 ∼= 838 955 ∼ 955 ∼= 937 956 ∼ 956 ∼= 956 957 ∼ 957 ∼= 957 958 ∼ 956 ∼= 956 959 ∼ 959 ∼= 959 960 ∼ 960 ∼= 960 961 ∼ 957 ∼= 957 962 ∼ 960 ∼= 960 963 ∼ 963 ∼= 963 964 ∼ 964 ∼= 739 965 ∼ 965 ∼= 965 966 ∼ 966 ∼= 966 967 ∼ 965 ∼= 965 968 ∼ 968 ∼= 968 969 ∼ 969 ∼= 969 970 ∼ 966 ∼= 966 971 ∼ 969 ∼= 969 972 ∼ 972 ∼= 739 973 ∼ 748 ∼= 748 974 ∼ 750 ∼= 750 975 ∼ 749 ∼= 749 976 ∼ 750 ∼= 750 977 ∼ 756 ∼= 748 978 ∼ 753 ∼= 753 979 ∼ 749 ∼= 749 980 ∼ 753 ∼= 753 981 ∼ 752 ∼= 752 982 ∼ 838 ∼= 838 983 ∼ 839 ∼= 821 984 ∼ 840 ∼= 840 985 ∼ 839 ∼= 821 986 ∼ 842 ∼= 838 987 ∼ 843 ∼= 843 988 ∼ 840 ∼= 840 989 ∼ 843 ∼= 843 990 ∼ 846 ∼= 846 991 ∼ 865 ∼= 820 992 ∼ 866 ∼= 866 993 ∼ 866 ∼= 866 994 ∼ 866 ∼= 866 995 ∼ 869 ∼= 869 996 ∼ 870 ∼= 870 997 ∼ 866 ∼= 866 998 ∼ 870 ∼= 870 999 ∼ 869 ∼= 869 1000 ∼ 820 ∼= 820 1001 ∼ 821 ∼= 821 1002 ∼ 821 ∼= 821 1003 ∼ 821 ∼= 821 1004 ∼ 824 ∼= 820 1005 ∼ 824 ∼= 820 1006 ∼ 821 ∼= 821 1007 ∼ 824 ∼= 820 1008 ∼ 824 ∼= 820 1009 ∼ 847 ∼= 847 1010 ∼ 848 ∼= 750 1011 ∼ 849 ∼= 849 1012 ∼ 848 ∼= 750 1013 ∼ 851 ∼= 847 1014 ∼ 852 ∼= 852 1015 ∼ 849 ∼= 849 1016 ∼ 852 ∼= 852 1017 ∼ 855 ∼= 847 1018 ∼ 874 ∼= 874 1019 ∼ 875 ∼= 875 1020 ∼ 876 ∼= 876 1021 ∼ 875 ∼= 875 1022 ∼ 878 ∼= 878 1023 ∼ 879 ∼= 879 1024 ∼ 876 ∼= 876 1025 ∼ 879 ∼= 879 1026 ∼ 882 ∼= 882 1027 ∼ 820 ∼= 820 1028 ∼ 821 ∼= 821 1029 ∼ 821 ∼= 821 1030 ∼ 821 ∼= 821 1031 ∼ 824 ∼= 820 1032 ∼ 824 ∼= 820 1033 ∼ 821 ∼= 821 1034 ∼ 824 ∼= 820 1035 ∼ 824 ∼= 820 1036 ∼ 856 ∼= 856 1037 ∼ 857 ∼= 857 1038 ∼ 858 ∼= 858 1039 ∼ 857 ∼= 857 1040 ∼ 860 ∼= 860 1041 ∼ 861 ∼= 861 1042 ∼ 858 ∼= 858 1043 ∼ 861 ∼= 861 1044 ∼ 864 ∼= 864 1045 ∼ 883 ∼= 883 22 Classification of groups generated by automata 1046 ∼ 884 ∼= 884 1047 ∼ 885 ∼= 885 1048 ∼ 884 ∼= 884 1049 ∼ 887 ∼= 887 1050 ∼ 888 ∼= 888 1051 ∼ 885 ∼= 885 1052 ∼ 888 ∼= 888 1053 ∼ 891 ∼= 891 1054 ∼ 802 ∼= 802 1055 ∼ 804 ∼= 731 1056 ∼ 803 ∼= 771 1057 ∼ 804 ∼= 731 1058 ∼ 810 ∼= 802 1059 ∼ 807 ∼= 771 1060 ∼ 803 ∼= 771 1061 ∼ 807 ∼= 771 1062 ∼ 806 ∼= 802 1063 ∼ 964 ∼= 739 1064 ∼ 966 ∼= 966 1065 ∼ 965 ∼= 965 1066 ∼ 966 ∼= 966 1067 ∼ 972 ∼= 739 1068 ∼ 969 ∼= 969 1069 ∼ 965 ∼= 965 1070 ∼ 969 ∼= 969 1071 ∼ 968 ∼= 968 1072 ∼ 883 ∼= 883 1073 ∼ 885 ∼= 885 1074 ∼ 884 ∼= 884 1075 ∼ 885 ∼= 885 1076 ∼ 891 ∼= 891 1077 ∼ 888 ∼= 888 1078 ∼ 884 ∼= 884 1079 ∼ 888 ∼= 888 1080 ∼ 887 ∼= 887 1081 ∼ 964 ∼= 739 1082 ∼ 966 ∼= 966 1083 ∼ 965 ∼= 965 1084 ∼ 966 ∼= 966 1085 ∼ 972 ∼= 739 1086 ∼ 969 ∼= 969 1087 ∼ 965 ∼= 965 1088 ∼ 969 ∼= 969 1089 ∼ 968 ∼= 968 1090 ∼ 1090 ∼= 1090 1091 ∼ 1091 ∼= 731 1092 ∼ 1091 ∼= 731 1093 ∼ 1091 ∼= 731 1094 ∼ 1094 ∼= 1090 1095 ∼ 1094 ∼= 1090 1096 ∼ 1091 ∼= 731 1097 ∼ 1094 ∼= 1090 1098 ∼ 1094 ∼= 1090 1099 ∼ 1090 ∼= 1090 1100 ∼ 1091 ∼= 731 1101 ∼ 1091 ∼= 731 1102 ∼ 1091 ∼= 731 1103 ∼ 1094 ∼= 1090 1104 ∼ 1094 ∼= 1090 1105 ∼ 1091 ∼= 731 1106 ∼ 1094 ∼= 1090 1107 ∼ 1094 ∼= 1090 1108 ∼ 883 ∼= 883 1109 ∼ 885 ∼= 885 1110 ∼ 884 ∼= 884 1111 ∼ 885 ∼= 885 1112 ∼ 891 ∼= 891 1113 ∼ 888 ∼= 888 1114 ∼ 884 ∼= 884 1115 ∼ 888 ∼= 888 1116 ∼ 887 ∼= 887 1117 ∼ 1090 ∼= 1090 1118 ∼ 1091 ∼= 731 1119 ∼ 1091 ∼= 731 1120 ∼ 1091 ∼= 731 1121 ∼ 1094 ∼= 1090 1122 ∼ 1094 ∼= 1090 1123 ∼ 1091 ∼= 731 1124 ∼ 1094 ∼= 1090 1125 ∼ 1094 ∼= 1090 1126 ∼ 1090 ∼= 1090 1127 ∼ 1091 ∼= 731 1128 ∼ 1091 ∼= 731 1129 ∼ 1091 ∼= 731 1130 ∼ 1094 ∼= 1090 1131 ∼ 1094 ∼= 1090 1132 ∼ 1091 ∼= 731 1133 ∼ 1094 ∼= 1090 1134 ∼ 1094 ∼= 1090 1135 ∼ 775 ∼= 775 1136 ∼ 777 ∼= 777 1137 ∼ 776 ∼= 776 1138 ∼ 777 ∼= 777 1139 ∼ 783 ∼= 775 1140 ∼ 780 ∼= 780 1141 ∼ 776 ∼= 776 1142 ∼ 780 ∼= 780 1143 ∼ 779 ∼= 779 1144 ∼ 955 ∼= 937 1145 ∼ 957 ∼= 957 1146 ∼ 956 ∼= 956 1147 ∼ 957 ∼= 957 1148 ∼ 963 ∼= 963 1149 ∼ 960 ∼= 960 1150 ∼ 956 ∼= 956 1151 ∼ 960 ∼= 960 1152 ∼ 959 ∼= 959 1153 ∼ 874 ∼= 874 1154 ∼ 876 ∼= 876 1155 ∼ 875 ∼= 875 1156 ∼ 876 ∼= 876 1157 ∼ 882 ∼= 882 1158 ∼ 879 ∼= 879 1159 ∼ 875 ∼= 875 1160 ∼ 879 ∼= 879 1161 ∼ 878 ∼= 878 1162 ∼ 937 ∼= 937 1163 ∼ 939 ∼= 939 1164 ∼ 938 ∼= 938 1165 ∼ 939 ∼= 939 1166 ∼ 945 ∼= 941 1167 ∼ 942 ∼= 942 1168 ∼ 938 ∼= 938 1169 ∼ 942 ∼= 942 1170 ∼ 941 ∼= 941 1171 ∼ 1090 ∼= 1090 1172 ∼ 1091 ∼= 731 1173 ∼ 1091 ∼= 731 1174 ∼ 1091 ∼= 731 1175 ∼ 1094 ∼= 1090 1176 ∼ 1094 ∼= 1090 1177 ∼ 1091 ∼= 731 1178 ∼ 1094 ∼= 1090 1179 ∼ 1094 ∼= 1090 1180 ∼ 1090 ∼= 1090 1181 ∼ 1091 ∼= 731 1182 ∼ 1091 ∼= 731 1183 ∼ 1091 ∼= 731 1184 ∼ 1094 ∼= 1090 1185 ∼ 1094 ∼= 1090 1186 ∼ 1091 ∼= 731 1187 ∼ 1094 ∼= 1090 1188 ∼ 1094 ∼= 1090 1189 ∼ 856 ∼= 856 1190 ∼ 858 ∼= 858 1191 ∼ 857 ∼= 857 1192 ∼ 858 ∼= 858 1193 ∼ 864 ∼= 864 1194 ∼ 861 ∼= 861 1195 ∼ 857 ∼= 857 1196 ∼ 861 ∼= 861 1197 ∼ 860 ∼= 860 1198 ∼ 1090 ∼= 1090 1199 ∼ 1091 ∼= 731 1200 ∼ 1091 ∼= 731 1201 ∼ 1091 ∼= 731 1202 ∼ 1094 ∼= 1090 1203 ∼ 1094 ∼= 1090 1204 ∼ 1091 ∼= 731 1205 ∼ 1094 ∼= 1090 1206 ∼ 1094 ∼= 1090 1207 ∼ 1090 ∼= 1090 1208 ∼ 1091 ∼= 731 1209 ∼ 1091 ∼= 731 1210 ∼ 1091 ∼= 731 1211 ∼ 1094 ∼= 1090 1212 ∼ 1094 ∼= 1090 1213 ∼ 1091 ∼= 731 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 1214 ∼ 1094 ∼= 1090 1215 ∼ 1094 ∼= 1090 1216 ∼ 739 ∼= 739 1217 ∼ 741 ∼= 741 1218 ∼ 740 ∼= 740 1219 ∼ 741 ∼= 741 1220 ∼ 747 ∼= 739 1221 ∼ 744 ∼= 744 1222 ∼ 740 ∼= 740 1223 ∼ 744 ∼= 744 1224 ∼ 743 ∼= 739 1225 ∼ 919 ∼= 820 1226 ∼ 920 ∼= 920 1227 ∼ 920 ∼= 920 1228 ∼ 920 ∼= 920 1229 ∼ 923 ∼= 923 1230 ∼ 924 ∼= 870 1231 ∼ 920 ∼= 920 1232 ∼ 924 ∼= 870 1233 ∼ 923 ∼= 923 1234 ∼ 838 ∼= 838 1235 ∼ 840 ∼= 840 1236 ∼ 839 ∼= 821 1237 ∼ 840 ∼= 840 1238 ∼ 846 ∼= 846 1239 ∼ 843 ∼= 843 1240 ∼ 839 ∼= 821 1241 ∼ 843 ∼= 843 1242 ∼ 842 ∼= 838 1243 ∼ 820 ∼= 820 1244 ∼ 821 ∼= 821 1245 ∼ 821 ∼= 821 1246 ∼ 821 ∼= 821 1247 ∼ 824 ∼= 820 1248 ∼ 824 ∼= 820 1249 ∼ 821 ∼= 821 1250 ∼ 824 ∼= 820 1251 ∼ 824 ∼= 820 1252 ∼ 928 ∼= 820 1253 ∼ 929 ∼= 929 1254 ∼ 930 ∼= 821 1255 ∼ 929 ∼= 929 1256 ∼ 932 ∼= 820 1257 ∼ 933 ∼= 849 1258 ∼ 930 ∼= 821 1259 ∼ 933 ∼= 849 1260 ∼ 936 ∼= 820 1261 ∼ 955 ∼= 937 1262 ∼ 956 ∼= 956 1263 ∼ 957 ∼= 957 1264 ∼ 956 ∼= 956 1265 ∼ 959 ∼= 959 1266 ∼ 960 ∼= 960 1267 ∼ 957 ∼= 957 1268 ∼ 960 ∼= 960 1269 ∼ 963 ∼= 963 1270 ∼ 820 ∼= 820 1271 ∼ 821 ∼= 821 1272 ∼ 821 ∼= 821 1273 ∼ 821 ∼= 821 1274 ∼ 824 ∼= 820 1275 ∼ 824 ∼= 820 1276 ∼ 821 ∼= 821 1277 ∼ 824 ∼= 820 1278 ∼ 824 ∼= 820 1279 ∼ 937 ∼= 937 1280 ∼ 938 ∼= 938 1281 ∼ 939 ∼= 939 1282 ∼ 938 ∼= 938 1283 ∼ 941 ∼= 941 1284 ∼ 942 ∼= 942 1285 ∼ 939 ∼= 939 1286 ∼ 942 ∼= 942 1287 ∼ 945 ∼= 941 1288 ∼ 964 ∼= 739 1289 ∼ 965 ∼= 965 1290 ∼ 966 ∼= 966 1291 ∼ 965 ∼= 965 1292 ∼ 968 ∼= 968 1293 ∼ 969 ∼= 969 1294 ∼ 966 ∼= 966 1295 ∼ 969 ∼= 969 1296 ∼ 972 ∼= 739 1297 ∼ 775 ∼= 775 1298 ∼ 777 ∼= 777 1299 ∼ 776 ∼= 776 1300 ∼ 777 ∼= 777 1301 ∼ 783 ∼= 775 1302 ∼ 780 ∼= 780 1303 ∼ 776 ∼= 776 1304 ∼ 780 ∼= 780 1305 ∼ 779 ∼= 779 1306 ∼ 937 ∼= 937 1307 ∼ 939 ∼= 939 1308 ∼ 938 ∼= 938 1309 ∼ 939 ∼= 939 1310 ∼ 945 ∼= 941 1311 ∼ 942 ∼= 942 1312 ∼ 938 ∼= 938 1313 ∼ 942 ∼= 942 1314 ∼ 941 ∼= 941 1315 ∼ 856 ∼= 856 1316 ∼ 858 ∼= 858 1317 ∼ 857 ∼= 857 1318 ∼ 858 ∼= 858 1319 ∼ 864 ∼= 864 1320 ∼ 861 ∼= 861 1321 ∼ 857 ∼= 857 1322 ∼ 861 ∼= 861 1323 ∼ 860 ∼= 860 1324 ∼ 955 ∼= 937 1325 ∼ 957 ∼= 957 1326 ∼ 956 ∼= 956 1327 ∼ 957 ∼= 957 1328 ∼ 963 ∼= 963 1329 ∼ 960 ∼= 960 1330 ∼ 956 ∼= 956 1331 ∼ 960 ∼= 960 1332 ∼ 959 ∼= 959 1333 ∼ 1090 ∼= 1090 1334 ∼ 1091 ∼= 731 1335 ∼ 1091 ∼= 731 1336 ∼ 1091 ∼= 731 1337 ∼ 1094 ∼= 1090 1338 ∼ 1094 ∼= 1090 1339 ∼ 1091 ∼= 731 1340 ∼ 1094 ∼= 1090 1341 ∼ 1094 ∼= 1090 1342 ∼ 1090 ∼= 1090 1343 ∼ 1091 ∼= 731 1344 ∼ 1091 ∼= 731 1345 ∼ 1091 ∼= 731 1346 ∼ 1094 ∼= 1090 1347 ∼ 1094 ∼= 1090 1348 ∼ 1091 ∼= 731 1349 ∼ 1094 ∼= 1090 1350 ∼ 1094 ∼= 1090 1351 ∼ 874 ∼= 874 1352 ∼ 876 ∼= 876 1353 ∼ 875 ∼= 875 1354 ∼ 876 ∼= 876 1355 ∼ 882 ∼= 882 1356 ∼ 879 ∼= 879 1357 ∼ 875 ∼= 875 1358 ∼ 879 ∼= 879 1359 ∼ 878 ∼= 878 1360 ∼ 1090 ∼= 1090 1361 ∼ 1091 ∼= 731 1362 ∼ 1091 ∼= 731 1363 ∼ 1091 ∼= 731 1364 ∼ 1094 ∼= 1090 1365 ∼ 1094 ∼= 1090 1366 ∼ 1091 ∼= 731 1367 ∼ 1094 ∼= 1090 1368 ∼ 1094 ∼= 1090 1369 ∼ 1090 ∼= 1090 1370 ∼ 1091 ∼= 731 1371 ∼ 1091 ∼= 731 1372 ∼ 1091 ∼= 731 1373 ∼ 1094 ∼= 1090 1374 ∼ 1094 ∼= 1090 1375 ∼ 1091 ∼= 731 1376 ∼ 1094 ∼= 1090 1377 ∼ 1094 ∼= 1090 1378 ∼ 766 ∼= 730 1379 ∼ 768 ∼= 731 1380 ∼ 767 ∼= 731 1381 ∼ 768 ∼= 731 24 Classification of groups generated by automata 1382 ∼ 774 ∼= 730 1383 ∼ 771 ∼= 771 1384 ∼ 767 ∼= 731 1385 ∼ 771 ∼= 771 1386 ∼ 770 ∼= 730 1387 ∼ 928 ∼= 820 1388 ∼ 930 ∼= 821 1389 ∼ 929 ∼= 929 1390 ∼ 930 ∼= 821 1391 ∼ 936 ∼= 820 1392 ∼ 933 ∼= 849 1393 ∼ 929 ∼= 929 1394 ∼ 933 ∼= 849 1395 ∼ 932 ∼= 820 1396 ∼ 847 ∼= 847 1397 ∼ 849 ∼= 849 1398 ∼ 848 ∼= 750 1399 ∼ 849 ∼= 849 1400 ∼ 855 ∼= 847 1401 ∼ 852 ∼= 852 1402 ∼ 848 ∼= 750 1403 ∼ 852 ∼= 852 1404 ∼ 851 ∼= 847 1405 ∼ 928 ∼= 820 1406 ∼ 930 ∼= 821 1407 ∼ 929 ∼= 929 1408 ∼ 930 ∼= 821 1409 ∼ 936 ∼= 820 1410 ∼ 933 ∼= 849 1411 ∼ 929 ∼= 929 1412 ∼ 933 ∼= 849 1413 ∼ 932 ∼= 820 1414 ∼ 1090 ∼= 1090 1415 ∼ 1091 ∼= 731 1416 ∼ 1091 ∼= 731 1417 ∼ 1091 ∼= 731 1418 ∼ 1094 ∼= 1090 1419 ∼ 1094 ∼= 1090 1420 ∼ 1091 ∼= 731 1421 ∼ 1094 ∼= 1090 1422 ∼ 1094 ∼= 1090 1423 ∼ 1090 ∼= 1090 1424 ∼ 1091 ∼= 731 1425 ∼ 1091 ∼= 731 1426 ∼ 1091 ∼= 731 1427 ∼ 1094 ∼= 1090 1428 ∼ 1094 ∼= 1090 1429 ∼ 1091 ∼= 731 1430 ∼ 1094 ∼= 1090 1431 ∼ 1094 ∼= 1090 1432 ∼ 847 ∼= 847 1433 ∼ 849 ∼= 849 1434 ∼ 848 ∼= 750 1435 ∼ 849 ∼= 849 1436 ∼ 855 ∼= 847 1437 ∼ 852 ∼= 852 1438 ∼ 848 ∼= 750 1439 ∼ 852 ∼= 852 1440 ∼ 851 ∼= 847 1441 ∼ 1090 ∼= 1090 1442 ∼ 1091 ∼= 731 1443 ∼ 1091 ∼= 731 1444 ∼ 1091 ∼= 731 1445 ∼ 1094 ∼= 1090 1446 ∼ 1094 ∼= 1090 1447 ∼ 1091 ∼= 731 1448 ∼ 1094 ∼= 1090 1449 ∼ 1094 ∼= 1090 1450 ∼ 1090 ∼= 1090 1451 ∼ 1091 ∼= 731 1452 ∼ 1091 ∼= 731 1453 ∼ 1091 ∼= 731 1454 ∼ 1094 ∼= 1090 1455 ∼ 1094 ∼= 1090 1456 ∼ 1091 ∼= 731 1457 ∼ 1094 ∼= 1090 1458 ∼ 1094 ∼= 1090 1459 ∼ 1094 ∼= 1090 1460 ∼ 972 ∼= 739 1461 ∼ 1094 ∼= 1090 1462 ∼ 972 ∼= 739 1463 ∼ 810 ∼= 802 1464 ∼ 891 ∼= 891 1465 ∼ 1094 ∼= 1090 1466 ∼ 891 ∼= 891 1467 ∼ 1094 ∼= 1090 1468 ∼ 1091 ∼= 731 1469 ∼ 966 ∼= 966 1470 ∼ 1091 ∼= 731 1471 ∼ 966 ∼= 966 1472 ∼ 804 ∼= 731 1473 ∼ 885 ∼= 885 1474 ∼ 1091 ∼= 731 1475 ∼ 885 ∼= 885 1476 ∼ 1091 ∼= 731 1477 ∼ 1094 ∼= 1090 1478 ∼ 969 ∼= 969 1479 ∼ 1094 ∼= 1090 1480 ∼ 969 ∼= 969 1481 ∼ 807 ∼= 771 1482 ∼ 888 ∼= 888 1483 ∼ 1094 ∼= 1090 1484 ∼ 888 ∼= 888 1485 ∼ 1094 ∼= 1090 1486 ∼ 1091 ∼= 731 1487 ∼ 966 ∼= 966 1488 ∼ 1091 ∼= 731 1489 ∼ 966 ∼= 966 1490 ∼ 804 ∼= 731 1491 ∼ 885 ∼= 885 1492 ∼ 1091 ∼= 731 1493 ∼ 885 ∼= 885 1494 ∼ 1091 ∼= 731 1495 ∼ 1090 ∼= 1090 1496 ∼ 964 ∼= 739 1497 ∼ 1090 ∼= 1090 1498 ∼ 964 ∼= 739 1499 ∼ 802 ∼= 802 1500 ∼ 883 ∼= 883 1501 ∼ 1090 ∼= 1090 1502 ∼ 883 ∼= 883 1503 ∼ 1090 ∼= 1090 1504 ∼ 1091 ∼= 731 1505 ∼ 965 ∼= 965 1506 ∼ 1091 ∼= 731 1507 ∼ 965 ∼= 965 1508 ∼ 803 ∼= 771 1509 ∼ 884 ∼= 884 1510 ∼ 1091 ∼= 731 1511 ∼ 884 ∼= 884 1512 ∼ 1091 ∼= 731 1513 ∼ 1094 ∼= 1090 1514 ∼ 969 ∼= 969 1515 ∼ 1094 ∼= 1090 1516 ∼ 969 ∼= 969 1517 ∼ 807 ∼= 771 1518 ∼ 888 ∼= 888 1519 ∼ 1094 ∼= 1090 1520 ∼ 888 ∼= 888 1521 ∼ 1094 ∼= 1090 1522 ∼ 1091 ∼= 731 1523 ∼ 965 ∼= 965 1524 ∼ 1091 ∼= 731 1525 ∼ 965 ∼= 965 1526 ∼ 803 ∼= 771 1527 ∼ 884 ∼= 884 1528 ∼ 1091 ∼= 731 1529 ∼ 884 ∼= 884 1530 ∼ 1091 ∼= 731 1531 ∼ 1094 ∼= 1090 1532 ∼ 968 ∼= 968 1533 ∼ 1094 ∼= 1090 1534 ∼ 968 ∼= 968 1535 ∼ 806 ∼= 802 1536 ∼ 887 ∼= 887 1537 ∼ 1094 ∼= 1090 1538 ∼ 887 ∼= 887 1539 ∼ 1094 ∼= 1090 1540 ∼ 851 ∼= 847 1541 ∼ 824 ∼= 820 1542 ∼ 878 ∼= 878 1543 ∼ 842 ∼= 838 1544 ∼ 756 ∼= 748 1545 ∼ 869 ∼= 869 1546 ∼ 860 ∼= 860 1547 ∼ 824 ∼= 820 1548 ∼ 887 ∼= 887 1549 ∼ 848 ∼= 750 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 1550 ∼ 821 ∼= 821 1551 ∼ 875 ∼= 875 1552 ∼ 839 ∼= 821 1553 ∼ 750 ∼= 750 1554 ∼ 866 ∼= 866 1555 ∼ 857 ∼= 857 1556 ∼ 821 ∼= 821 1557 ∼ 884 ∼= 884 1558 ∼ 852 ∼= 852 1559 ∼ 824 ∼= 820 1560 ∼ 879 ∼= 879 1561 ∼ 843 ∼= 843 1562 ∼ 753 ∼= 753 1563 ∼ 870 ∼= 870 1564 ∼ 861 ∼= 861 1565 ∼ 824 ∼= 820 1566 ∼ 888 ∼= 888 1567 ∼ 848 ∼= 750 1568 ∼ 821 ∼= 821 1569 ∼ 875 ∼= 875 1570 ∼ 839 ∼= 821 1571 ∼ 750 ∼= 750 1572 ∼ 866 ∼= 866 1573 ∼ 857 ∼= 857 1574 ∼ 821 ∼= 821 1575 ∼ 884 ∼= 884 1576 ∼ 847 ∼= 847 1577 ∼ 820 ∼= 820 1578 ∼ 874 ∼= 874 1579 ∼ 838 ∼= 838 1580 ∼ 748 ∼= 748 1581 ∼ 865 ∼= 820 1582 ∼ 856 ∼= 856 1583 ∼ 820 ∼= 820 1584 ∼ 883 ∼= 883 1585 ∼ 849 ∼= 849 1586 ∼ 821 ∼= 821 1587 ∼ 876 ∼= 876 1588 ∼ 840 ∼= 840 1589 ∼ 749 ∼= 749 1590 ∼ 866 ∼= 866 1591 ∼ 858 ∼= 858 1592 ∼ 821 ∼= 821 1593 ∼ 885 ∼= 885 1594 ∼ 852 ∼= 852 1595 ∼ 824 ∼= 820 1596 ∼ 879 ∼= 879 1597 ∼ 843 ∼= 843 1598 ∼ 753 ∼= 753 1599 ∼ 870 ∼= 870 1600 ∼ 861 ∼= 861 1601 ∼ 824 ∼= 820 1602 ∼ 888 ∼= 888 1603 ∼ 849 ∼= 849 1604 ∼ 821 ∼= 821 1605 ∼ 876 ∼= 876 1606 ∼ 840 ∼= 840 1607 ∼ 749 ∼= 749 1608 ∼ 866 ∼= 866 1609 ∼ 858 ∼= 858 1610 ∼ 821 ∼= 821 1611 ∼ 885 ∼= 885 1612 ∼ 855 ∼= 847 1613 ∼ 824 ∼= 820 1614 ∼ 882 ∼= 882 1615 ∼ 846 ∼= 846 1616 ∼ 752 ∼= 752 1617 ∼ 869 ∼= 869 1618 ∼ 864 ∼= 864 1619 ∼ 824 ∼= 820 1620 ∼ 891 ∼= 891 1621 ∼ 1094 ∼= 1090 1622 ∼ 945 ∼= 941 1623 ∼ 1094 ∼= 1090 1624 ∼ 963 ∼= 963 1625 ∼ 783 ∼= 775 1626 ∼ 882 ∼= 882 1627 ∼ 1094 ∼= 1090 1628 ∼ 864 ∼= 864 1629 ∼ 1094 ∼= 1090 1630 ∼ 1091 ∼= 731 1631 ∼ 939 ∼= 939 1632 ∼ 1091 ∼= 731 1633 ∼ 957 ∼= 957 1634 ∼ 777 ∼= 777 1635 ∼ 876 ∼= 876 1636 ∼ 1091 ∼= 731 1637 ∼ 858 ∼= 858 1638 ∼ 1091 ∼= 731 1639 ∼ 1094 ∼= 1090 1640 ∼ 942 ∼= 942 1641 ∼ 1094 ∼= 1090 1642 ∼ 960 ∼= 960 1643 ∼ 780 ∼= 780 1644 ∼ 879 ∼= 879 1645 ∼ 1094 ∼= 1090 1646 ∼ 861 ∼= 861 1647 ∼ 1094 ∼= 1090 1648 ∼ 1091 ∼= 731 1649 ∼ 939 ∼= 939 1650 ∼ 1091 ∼= 731 1651 ∼ 957 ∼= 957 1652 ∼ 777 ∼= 777 1653 ∼ 876 ∼= 876 1654 ∼ 1091 ∼= 731 1655 ∼ 858 ∼= 858 1656 ∼ 1091 ∼= 731 1657 ∼ 1090 ∼= 1090 1658 ∼ 937 ∼= 937 1659 ∼ 1090 ∼= 1090 1660 ∼ 955 ∼= 937 1661 ∼ 775 ∼= 775 1662 ∼ 874 ∼= 874 1663 ∼ 1090 ∼= 1090 1664 ∼ 856 ∼= 856 1665 ∼ 1090 ∼= 1090 1666 ∼ 1091 ∼= 731 1667 ∼ 938 ∼= 938 1668 ∼ 1091 ∼= 731 1669 ∼ 956 ∼= 956 1670 ∼ 776 ∼= 776 1671 ∼ 875 ∼= 875 1672 ∼ 1091 ∼= 731 1673 ∼ 857 ∼= 857 1674 ∼ 1091 ∼= 731 1675 ∼ 1094 ∼= 1090 1676 ∼ 942 ∼= 942 1677 ∼ 1094 ∼= 1090 1678 ∼ 960 ∼= 960 1679 ∼ 780 ∼= 780 1680 ∼ 879 ∼= 879 1681 ∼ 1094 ∼= 1090 1682 ∼ 861 ∼= 861 1683 ∼ 1094 ∼= 1090 1684 ∼ 1091 ∼= 731 1685 ∼ 938 ∼= 938 1686 ∼ 1091 ∼= 731 1687 ∼ 956 ∼= 956 1688 ∼ 776 ∼= 776 1689 ∼ 875 ∼= 875 1690 ∼ 1091 ∼= 731 1691 ∼ 857 ∼= 857 1692 ∼ 1091 ∼= 731 1693 ∼ 1094 ∼= 1090 1694 ∼ 941 ∼= 941 1695 ∼ 1094 ∼= 1090 1696 ∼ 959 ∼= 959 1697 ∼ 779 ∼= 779 1698 ∼ 878 ∼= 878 1699 ∼ 1094 ∼= 1090 1700 ∼ 860 ∼= 860 1701 ∼ 1094 ∼= 1090 1702 ∼ 851 ∼= 847 1703 ∼ 842 ∼= 838 1704 ∼ 860 ∼= 860 1705 ∼ 824 ∼= 820 1706 ∼ 756 ∼= 748 1707 ∼ 824 ∼= 820 1708 ∼ 878 ∼= 878 1709 ∼ 869 ∼= 869 1710 ∼ 887 ∼= 887 1711 ∼ 848 ∼= 750 1712 ∼ 839 ∼= 821 1713 ∼ 857 ∼= 857 1714 ∼ 821 ∼= 821 1715 ∼ 750 ∼= 750 1716 ∼ 821 ∼= 821 1717 ∼ 875 ∼= 875 26 Classification of groups generated by automata 1718 ∼ 866 ∼= 866 1719 ∼ 884 ∼= 884 1720 ∼ 852 ∼= 852 1721 ∼ 843 ∼= 843 1722 ∼ 861 ∼= 861 1723 ∼ 824 ∼= 820 1724 ∼ 753 ∼= 753 1725 ∼ 824 ∼= 820 1726 ∼ 879 ∼= 879 1727 ∼ 870 ∼= 870 1728 ∼ 888 ∼= 888 1729 ∼ 848 ∼= 750 1730 ∼ 839 ∼= 821 1731 ∼ 857 ∼= 857 1732 ∼ 821 ∼= 821 1733 ∼ 750 ∼= 750 1734 ∼ 821 ∼= 821 1735 ∼ 875 ∼= 875 1736 ∼ 866 ∼= 866 1737 ∼ 884 ∼= 884 1738 ∼ 847 ∼= 847 1739 ∼ 838 ∼= 838 1740 ∼ 856 ∼= 856 1741 ∼ 820 ∼= 820 1742 ∼ 748 ∼= 748 1743 ∼ 820 ∼= 820 1744 ∼ 874 ∼= 874 1745 ∼ 865 ∼= 820 1746 ∼ 883 ∼= 883 1747 ∼ 849 ∼= 849 1748 ∼ 840 ∼= 840 1749 ∼ 858 ∼= 858 1750 ∼ 821 ∼= 821 1751 ∼ 749 ∼= 749 1752 ∼ 821 ∼= 821 1753 ∼ 876 ∼= 876 1754 ∼ 866 ∼= 866 1755 ∼ 885 ∼= 885 1756 ∼ 852 ∼= 852 1757 ∼ 843 ∼= 843 1758 ∼ 861 ∼= 861 1759 ∼ 824 ∼= 820 1760 ∼ 753 ∼= 753 1761 ∼ 824 ∼= 820 1762 ∼ 879 ∼= 879 1763 ∼ 870 ∼= 870 1764 ∼ 888 ∼= 888 1765 ∼ 849 ∼= 849 1766 ∼ 840 ∼= 840 1767 ∼ 858 ∼= 858 1768 ∼ 821 ∼= 821 1769 ∼ 749 ∼= 749 1770 ∼ 821 ∼= 821 1771 ∼ 876 ∼= 876 1772 ∼ 866 ∼= 866 1773 ∼ 885 ∼= 885 1774 ∼ 855 ∼= 847 1775 ∼ 846 ∼= 846 1776 ∼ 864 ∼= 864 1777 ∼ 824 ∼= 820 1778 ∼ 752 ∼= 752 1779 ∼ 824 ∼= 820 1780 ∼ 882 ∼= 882 1781 ∼ 869 ∼= 869 1782 ∼ 891 ∼= 891 1783 ∼ 770 ∼= 730 1784 ∼ 743 ∼= 739 1785 ∼ 779 ∼= 779 1786 ∼ 743 ∼= 739 1787 ∼ 734 ∼= 730 1788 ∼ 752 ∼= 752 1789 ∼ 779 ∼= 779 1790 ∼ 752 ∼= 752 1791 ∼ 806 ∼= 802 1792 ∼ 767 ∼= 731 1793 ∼ 740 ∼= 740 1794 ∼ 776 ∼= 776 1795 ∼ 740 ∼= 740 1796 ∼ 731 ∼= 731 1797 ∼ 749 ∼= 749 1798 ∼ 776 ∼= 776 1799 ∼ 749 ∼= 749 1800 ∼ 803 ∼= 771 1801 ∼ 771 ∼= 771 1802 ∼ 744 ∼= 744 1803 ∼ 780 ∼= 780 1804 ∼ 744 ∼= 744 1805 ∼ 734 ∼= 730 1806 ∼ 753 ∼= 753 1807 ∼ 780 ∼= 780 1808 ∼ 753 ∼= 753 1809 ∼ 807 ∼= 771 1810 ∼ 767 ∼= 731 1811 ∼ 740 ∼= 740 1812 ∼ 776 ∼= 776 1813 ∼ 740 ∼= 740 1814 ∼ 731 ∼= 731 1815 ∼ 749 ∼= 749 1816 ∼ 776 ∼= 776 1817 ∼ 749 ∼= 749 1818 ∼ 803 ∼= 771 1819 ∼ 766 ∼= 730 1820 ∼ 739 ∼= 739 1821 ∼ 775 ∼= 775 1822 ∼ 739 ∼= 739 1823 ∼ 730 ∼= 730 1824 ∼ 748 ∼= 748 1825 ∼ 775 ∼= 775 1826 ∼ 748 ∼= 748 1827 ∼ 802 ∼= 802 1828 ∼ 768 ∼= 731 1829 ∼ 741 ∼= 741 1830 ∼ 777 ∼= 777 1831 ∼ 741 ∼= 741 1832 ∼ 731 ∼= 731 1833 ∼ 750 ∼= 750 1834 ∼ 777 ∼= 777 1835 ∼ 750 ∼= 750 1836 ∼ 804 ∼= 731 1837 ∼ 771 ∼= 771 1838 ∼ 744 ∼= 744 1839 ∼ 780 ∼= 780 1840 ∼ 744 ∼= 744 1841 ∼ 734 ∼= 730 1842 ∼ 753 ∼= 753 1843 ∼ 780 ∼= 780 1844 ∼ 753 ∼= 753 1845 ∼ 807 ∼= 771 1846 ∼ 768 ∼= 731 1847 ∼ 741 ∼= 741 1848 ∼ 777 ∼= 777 1849 ∼ 741 ∼= 741 1850 ∼ 731 ∼= 731 1851 ∼ 750 ∼= 750 1852 ∼ 777 ∼= 777 1853 ∼ 750 ∼= 750 1854 ∼ 804 ∼= 731 1855 ∼ 774 ∼= 730 1856 ∼ 747 ∼= 739 1857 ∼ 783 ∼= 775 1858 ∼ 747 ∼= 739 1859 ∼ 734 ∼= 730 1860 ∼ 756 ∼= 748 1861 ∼ 783 ∼= 775 1862 ∼ 756 ∼= 748 1863 ∼ 810 ∼= 802 1864 ∼ 932 ∼= 820 1865 ∼ 923 ∼= 923 1866 ∼ 941 ∼= 941 1867 ∼ 824 ∼= 820 1868 ∼ 747 ∼= 739 1869 ∼ 824 ∼= 820 1870 ∼ 959 ∼= 959 1871 ∼ 846 ∼= 846 1872 ∼ 968 ∼= 968 1873 ∼ 929 ∼= 929 1874 ∼ 920 ∼= 920 1875 ∼ 938 ∼= 938 1876 ∼ 821 ∼= 821 1877 ∼ 741 ∼= 741 1878 ∼ 821 ∼= 821 1879 ∼ 956 ∼= 956 1880 ∼ 840 ∼= 840 1881 ∼ 965 ∼= 965 1882 ∼ 933 ∼= 849 1883 ∼ 924 ∼= 870 1884 ∼ 942 ∼= 942 1885 ∼ 824 ∼= 820 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 1886 ∼ 744 ∼= 744 1887 ∼ 824 ∼= 820 1888 ∼ 960 ∼= 960 1889 ∼ 843 ∼= 843 1890 ∼ 969 ∼= 969 1891 ∼ 929 ∼= 929 1892 ∼ 920 ∼= 920 1893 ∼ 938 ∼= 938 1894 ∼ 821 ∼= 821 1895 ∼ 741 ∼= 741 1896 ∼ 821 ∼= 821 1897 ∼ 956 ∼= 956 1898 ∼ 840 ∼= 840 1899 ∼ 965 ∼= 965 1900 ∼ 928 ∼= 820 1901 ∼ 919 ∼= 820 1902 ∼ 937 ∼= 937 1903 ∼ 820 ∼= 820 1904 ∼ 739 ∼= 739 1905 ∼ 820 ∼= 820 1906 ∼ 955 ∼= 937 1907 ∼ 838 ∼= 838 1908 ∼ 964 ∼= 739 1909 ∼ 930 ∼= 821 1910 ∼ 920 ∼= 920 1911 ∼ 939 ∼= 939 1912 ∼ 821 ∼= 821 1913 ∼ 740 ∼= 740 1914 ∼ 821 ∼= 821 1915 ∼ 957 ∼= 957 1916 ∼ 839 ∼= 821 1917 ∼ 966 ∼= 966 1918 ∼ 933 ∼= 849 1919 ∼ 924 ∼= 870 1920 ∼ 942 ∼= 942 1921 ∼ 824 ∼= 820 1922 ∼ 744 ∼= 744 1923 ∼ 824 ∼= 820 1924 ∼ 960 ∼= 960 1925 ∼ 843 ∼= 843 1926 ∼ 969 ∼= 969 1927 ∼ 930 ∼= 821 1928 ∼ 920 ∼= 920 1929 ∼ 939 ∼= 939 1930 ∼ 821 ∼= 821 1931 ∼ 740 ∼= 740 1932 ∼ 821 ∼= 821 1933 ∼ 957 ∼= 957 1934 ∼ 839 ∼= 821 1935 ∼ 966 ∼= 966 1936 ∼ 936 ∼= 820 1937 ∼ 923 ∼= 923 1938 ∼ 945 ∼= 941 1939 ∼ 824 ∼= 820 1940 ∼ 743 ∼= 739 1941 ∼ 824 ∼= 820 1942 ∼ 963 ∼= 963 1943 ∼ 842 ∼= 838 1944 ∼ 972 ∼= 739 1945 ∼ 1094 ∼= 1090 1946 ∼ 963 ∼= 963 1947 ∼ 1094 ∼= 1090 1948 ∼ 945 ∼= 941 1949 ∼ 783 ∼= 775 1950 ∼ 864 ∼= 864 1951 ∼ 1094 ∼= 1090 1952 ∼ 882 ∼= 882 1953 ∼ 1094 ∼= 1090 1954 ∼ 1091 ∼= 731 1955 ∼ 957 ∼= 957 1956 ∼ 1091 ∼= 731 1957 ∼ 939 ∼= 939 1958 ∼ 777 ∼= 777 1959 ∼ 858 ∼= 858 1960 ∼ 1091 ∼= 731 1961 ∼ 876 ∼= 876 1962 ∼ 1091 ∼= 731 1963 ∼ 1094 ∼= 1090 1964 ∼ 960 ∼= 960 1965 ∼ 1094 ∼= 1090 1966 ∼ 942 ∼= 942 1967 ∼ 780 ∼= 780 1968 ∼ 861 ∼= 861 1969 ∼ 1094 ∼= 1090 1970 ∼ 879 ∼= 879 1971 ∼ 1094 ∼= 1090 1972 ∼ 1091 ∼= 731 1973 ∼ 957 ∼= 957 1974 ∼ 1091 ∼= 731 1975 ∼ 939 ∼= 939 1976 ∼ 777 ∼= 777 1977 ∼ 858 ∼= 858 1978 ∼ 1091 ∼= 731 1979 ∼ 876 ∼= 876 1980 ∼ 1091 ∼= 731 1981 ∼ 1090 ∼= 1090 1982 ∼ 955 ∼= 937 1983 ∼ 1090 ∼= 1090 1984 ∼ 937 ∼= 937 1985 ∼ 775 ∼= 775 1986 ∼ 856 ∼= 856 1987 ∼ 1090 ∼= 1090 1988 ∼ 874 ∼= 874 1989 ∼ 1090 ∼= 1090 1990 ∼ 1091 ∼= 731 1991 ∼ 956 ∼= 956 1992 ∼ 1091 ∼= 731 1993 ∼ 938 ∼= 938 1994 ∼ 776 ∼= 776 1995 ∼ 857 ∼= 857 1996 ∼ 1091 ∼= 731 1997 ∼ 875 ∼= 875 1998 ∼ 1091 ∼= 731 1999 ∼ 1094 ∼= 1090 2000 ∼ 960 ∼= 960 2001 ∼ 1094 ∼= 1090 2002 ∼ 942 ∼= 942 2003 ∼ 780 ∼= 780 2004 ∼ 861 ∼= 861 2005 ∼ 1094 ∼= 1090 2006 ∼ 879 ∼= 879 2007 ∼ 1094 ∼= 1090 2008 ∼ 1091 ∼= 731 2009 ∼ 956 ∼= 956 2010 ∼ 1091 ∼= 731 2011 ∼ 938 ∼= 938 2012 ∼ 776 ∼= 776 2013 ∼ 857 ∼= 857 2014 ∼ 1091 ∼= 731 2015 ∼ 875 ∼= 875 2016 ∼ 1091 ∼= 731 2017 ∼ 1094 ∼= 1090 2018 ∼ 959 ∼= 959 2019 ∼ 1094 ∼= 1090 2020 ∼ 941 ∼= 941 2021 ∼ 779 ∼= 779 2022 ∼ 860 ∼= 860 2023 ∼ 1094 ∼= 1090 2024 ∼ 878 ∼= 878 2025 ∼ 1094 ∼= 1090 2026 ∼ 932 ∼= 820 2027 ∼ 824 ∼= 820 2028 ∼ 959 ∼= 959 2029 ∼ 923 ∼= 923 2030 ∼ 747 ∼= 739 2031 ∼ 846 ∼= 846 2032 ∼ 941 ∼= 941 2033 ∼ 824 ∼= 820 2034 ∼ 968 ∼= 968 2035 ∼ 929 ∼= 929 2036 ∼ 821 ∼= 821 2037 ∼ 956 ∼= 956 2038 ∼ 920 ∼= 920 2039 ∼ 741 ∼= 741 2040 ∼ 840 ∼= 840 2041 ∼ 938 ∼= 938 2042 ∼ 821 ∼= 821 2043 ∼ 965 ∼= 965 2044 ∼ 933 ∼= 849 2045 ∼ 824 ∼= 820 2046 ∼ 960 ∼= 960 2047 ∼ 924 ∼= 870 2048 ∼ 744 ∼= 744 2049 ∼ 843 ∼= 843 2050 ∼ 942 ∼= 942 2051 ∼ 824 ∼= 820 2052 ∼ 969 ∼= 969 2053 ∼ 929 ∼= 929 28 Classification of groups generated by automata 2054 ∼ 821 ∼= 821 2055 ∼ 956 ∼= 956 2056 ∼ 920 ∼= 920 2057 ∼ 741 ∼= 741 2058 ∼ 840 ∼= 840 2059 ∼ 938 ∼= 938 2060 ∼ 821 ∼= 821 2061 ∼ 965 ∼= 965 2062 ∼ 928 ∼= 820 2063 ∼ 820 ∼= 820 2064 ∼ 955 ∼= 937 2065 ∼ 919 ∼= 820 2066 ∼ 739 ∼= 739 2067 ∼ 838 ∼= 838 2068 ∼ 937 ∼= 937 2069 ∼ 820 ∼= 820 2070 ∼ 964 ∼= 739 2071 ∼ 930 ∼= 821 2072 ∼ 821 ∼= 821 2073 ∼ 957 ∼= 957 2074 ∼ 920 ∼= 920 2075 ∼ 740 ∼= 740 2076 ∼ 839 ∼= 821 2077 ∼ 939 ∼= 939 2078 ∼ 821 ∼= 821 2079 ∼ 966 ∼= 966 2080 ∼ 933 ∼= 849 2081 ∼ 824 ∼= 820 2082 ∼ 960 ∼= 960 2083 ∼ 924 ∼= 870 2084 ∼ 744 ∼= 744 2085 ∼ 843 ∼= 843 2086 ∼ 942 ∼= 942 2087 ∼ 824 ∼= 820 2088 ∼ 969 ∼= 969 2089 ∼ 930 ∼= 821 2090 ∼ 821 ∼= 821 2091 ∼ 957 ∼= 957 2092 ∼ 920 ∼= 920 2093 ∼ 740 ∼= 740 2094 ∼ 839 ∼= 821 2095 ∼ 939 ∼= 939 2096 ∼ 821 ∼= 821 2097 ∼ 966 ∼= 966 2098 ∼ 936 ∼= 820 2099 ∼ 824 ∼= 820 2100 ∼ 963 ∼= 963 2101 ∼ 923 ∼= 923 2102 ∼ 743 ∼= 739 2103 ∼ 842 ∼= 838 2104 ∼ 945 ∼= 941 2105 ∼ 824 ∼= 820 2106 ∼ 972 ∼= 739 2107 ∼ 1094 ∼= 1090 2108 ∼ 936 ∼= 820 2109 ∼ 1094 ∼= 1090 2110 ∼ 936 ∼= 820 2111 ∼ 774 ∼= 730 2112 ∼ 855 ∼= 847 2113 ∼ 1094 ∼= 1090 2114 ∼ 855 ∼= 847 2115 ∼ 1094 ∼= 1090 2116 ∼ 1091 ∼= 731 2117 ∼ 930 ∼= 821 2118 ∼ 1091 ∼= 731 2119 ∼ 930 ∼= 821 2120 ∼ 768 ∼= 731 2121 ∼ 849 ∼= 849 2122 ∼ 1091 ∼= 731 2123 ∼ 849 ∼= 849 2124 ∼ 1091 ∼= 731 2125 ∼ 1094 ∼= 1090 2126 ∼ 933 ∼= 849 2127 ∼ 1094 ∼= 1090 2128 ∼ 933 ∼= 849 2129 ∼ 771 ∼= 771 2130 ∼ 852 ∼= 852 2131 ∼ 1094 ∼= 1090 2132 ∼ 852 ∼= 852 2133 ∼ 1094 ∼= 1090 2134 ∼ 1091 ∼= 731 2135 ∼ 930 ∼= 821 2136 ∼ 1091 ∼= 731 2137 ∼ 930 ∼= 821 2138 ∼ 768 ∼= 731 2139 ∼ 849 ∼= 849 2140 ∼ 1091 ∼= 731 2141 ∼ 849 ∼= 849 2142 ∼ 1091 ∼= 731 2143 ∼ 1090 ∼= 1090 2144 ∼ 928 ∼= 820 2145 ∼ 1090 ∼= 1090 2146 ∼ 928 ∼= 820 2147 ∼ 766 ∼= 730 2148 ∼ 847 ∼= 847 2149 ∼ 1090 ∼= 1090 2150 ∼ 847 ∼= 847 2151 ∼ 1090 ∼= 1090 2152 ∼ 1091 ∼= 731 2153 ∼ 929 ∼= 929 2154 ∼ 1091 ∼= 731 2155 ∼ 929 ∼= 929 2156 ∼ 767 ∼= 731 2157 ∼ 848 ∼= 750 2158 ∼ 1091 ∼= 731 2159 ∼ 848 ∼= 750 2160 ∼ 1091 ∼= 731 2161 ∼ 1094 ∼= 1090 2162 ∼ 933 ∼= 849 2163 ∼ 1094 ∼= 1090 2164 ∼ 933 ∼= 849 2165 ∼ 771 ∼= 771 2166 ∼ 852 ∼= 852 2167 ∼ 1094 ∼= 1090 2168 ∼ 852 ∼= 852 2169 ∼ 1094 ∼= 1090 2170 ∼ 1091 ∼= 731 2171 ∼ 929 ∼= 929 2172 ∼ 1091 ∼= 731 2173 ∼ 929 ∼= 929 2174 ∼ 767 ∼= 731 2175 ∼ 848 ∼= 750 2176 ∼ 1091 ∼= 731 2177 ∼ 848 ∼= 750 2178 ∼ 1091 ∼= 731 2179 ∼ 1094 ∼= 1090 2180 ∼ 932 ∼= 820 2181 ∼ 1094 ∼= 1090 2182 ∼ 932 ∼= 820 2183 ∼ 770 ∼= 730 2184 ∼ 851 ∼= 847 2185 ∼ 1094 ∼= 1090 2186 ∼ 851 ∼= 847 2187 ∼ 1094 ∼= 1090 2188 ∼ 730 ∼= 730 2189 ∼ 730 ∼= 730 2190 ∼ 2190 ∼= 750 2191 ∼ 730 ∼= 730 2192 ∼ 730 ∼= 730 2193 ∼ 2193 ∼= 2193 2194 ∼ 2190 ∼= 750 2195 ∼ 2193 ∼= 2193 2196 ∼ 2196 ∼= 802 2197 ∼ 730 ∼= 730 2198 ∼ 730 ∼= 730 2199 ∼ 2199 ∼= 2199 2200 ∼ 730 ∼= 730 2201 ∼ 730 ∼= 730 2202 ∼ 2202 ∼= 2202 2203 ∼ 2203 ∼= 2203 2204 ∼ 2204 ∼= 2204 2205 ∼ 2205 ∼= 775 2206 ∼ 2206 ∼= 748 2207 ∼ 2207 ∼= 2207 2208 ∼ 731 ∼= 731 2209 ∼ 2209 ∼= 2209 2210 ∼ 2210 ∼= 2210 2211 ∼ 731 ∼= 731 2212 ∼ 2212 ∼= 2212 2213 ∼ 2213 ∼= 2213 2214 ∼ 2214 ∼= 748 2215 ∼ 730 ∼= 730 2216 ∼ 730 ∼= 730 2217 ∼ 2203 ∼= 2203 2218 ∼ 730 ∼= 730 2219 ∼ 730 ∼= 730 2220 ∼ 2204 ∼= 2204 2221 ∼ 2199 ∼= 2199 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 2222 ∼ 2202 ∼= 2202 2223 ∼ 2205 ∼= 775 2224 ∼ 730 ∼= 730 2225 ∼ 730 ∼= 730 2226 ∼ 2226 ∼= 820 2227 ∼ 730 ∼= 730 2228 ∼ 730 ∼= 730 2229 ∼ 2229 ∼= 2229 2230 ∼ 2226 ∼= 820 2231 ∼ 2229 ∼= 2229 2232 ∼ 2232 ∼= 730 2233 ∼ 2233 ∼= 2233 2234 ∼ 2234 ∼= 2234 2235 ∼ 731 ∼= 731 2236 ∼ 2236 ∼= 2236 2237 ∼ 2237 ∼= 2237 2238 ∼ 731 ∼= 731 2239 ∼ 2239 ∼= 2239 2240 ∼ 2240 ∼= 2240 2241 ∼ 2241 ∼= 739 2242 ∼ 2206 ∼= 748 2243 ∼ 2209 ∼= 2209 2244 ∼ 2212 ∼= 2212 2245 ∼ 2207 ∼= 2207 2246 ∼ 2210 ∼= 2210 2247 ∼ 2213 ∼= 2213 2248 ∼ 731 ∼= 731 2249 ∼ 731 ∼= 731 2250 ∼ 2214 ∼= 748 2251 ∼ 2233 ∼= 2233 2252 ∼ 2236 ∼= 2236 2253 ∼ 2239 ∼= 2239 2254 ∼ 2234 ∼= 2234 2255 ∼ 2237 ∼= 2237 2256 ∼ 2240 ∼= 2240 2257 ∼ 731 ∼= 731 2258 ∼ 731 ∼= 731 2259 ∼ 2241 ∼= 739 2260 ∼ 2260 ∼= 802 2261 ∼ 2261 ∼= 2261 2262 ∼ 2262 ∼= 750 2263 ∼ 2261 ∼= 2261 2264 ∼ 2264 ∼= 730 2265 ∼ 2265 ∼= 2265 2266 ∼ 2262 ∼= 750 2267 ∼ 2265 ∼= 2265 2268 ∼ 734 ∼= 730 2269 ∼ 730 ∼= 730 2270 ∼ 730 ∼= 730 2271 ∼ 2271 ∼= 2271 2272 ∼ 730 ∼= 730 2273 ∼ 730 ∼= 730 2274 ∼ 2274 ∼= 2274 2275 ∼ 2271 ∼= 2271 2276 ∼ 2274 ∼= 2274 2277 ∼ 2277 ∼= 2277 2278 ∼ 730 ∼= 730 2279 ∼ 730 ∼= 730 2280 ∼ 2280 ∼= 2280 2281 ∼ 730 ∼= 730 2282 ∼ 730 ∼= 730 2283 ∼ 2283 ∼= 2283 2284 ∼ 2284 ∼= 2284 2285 ∼ 2285 ∼= 2285 2286 ∼ 2286 ∼= 2286 2287 ∼ 2287 ∼= 2287 2288 ∼ 2285 ∼= 2285 2289 ∼ 731 ∼= 731 2290 ∼ 2283 ∼= 2283 2291 ∼ 2274 ∼= 2274 2292 ∼ 731 ∼= 731 2293 ∼ 2293 ∼= 2293 2294 ∼ 2294 ∼= 2294 2295 ∼ 2295 ∼= 2295 2296 ∼ 730 ∼= 730 2297 ∼ 730 ∼= 730 2298 ∼ 2284 ∼= 2284 2299 ∼ 730 ∼= 730 2300 ∼ 730 ∼= 730 2301 ∼ 2285 ∼= 2285 2302 ∼ 2280 ∼= 2280 2303 ∼ 2283 ∼= 2283 2304 ∼ 2286 ∼= 2286 2305 ∼ 730 ∼= 730 2306 ∼ 730 ∼= 730 2307 ∼ 2307 ∼= 2307 2308 ∼ 730 ∼= 730 2309 ∼ 730 ∼= 730 2310 ∼ 2287 ∼= 2287 2311 ∼ 2307 ∼= 2307 2312 ∼ 2287 ∼= 2287 2313 ∼ 2313 ∼= 2277 2314 ∼ 2307 ∼= 2307 2315 ∼ 2284 ∼= 2284 2316 ∼ 731 ∼= 731 2317 ∼ 2280 ∼= 2280 2318 ∼ 2271 ∼= 2271 2319 ∼ 731 ∼= 731 2320 ∼ 2320 ∼= 2294 2321 ∼ 2293 ∼= 2293 2322 ∼ 2322 ∼= 2322 2323 ∼ 2287 ∼= 2287 2324 ∼ 2283 ∼= 2283 2325 ∼ 2293 ∼= 2293 2326 ∼ 2285 ∼= 2285 2327 ∼ 2274 ∼= 2274 2328 ∼ 2294 ∼= 2294 2329 ∼ 731 ∼= 731 2330 ∼ 731 ∼= 731 2331 ∼ 2295 ∼= 2295 2332 ∼ 2307 ∼= 2307 2333 ∼ 2280 ∼= 2280 2334 ∼ 2320 ∼= 2294 2335 ∼ 2284 ∼= 2284 2336 ∼ 2271 ∼= 2271 2337 ∼ 2293 ∼= 2293 2338 ∼ 731 ∼= 731 2339 ∼ 731 ∼= 731 2340 ∼ 2322 ∼= 2322 2341 ∼ 2313 ∼= 2277 2342 ∼ 2286 ∼= 2286 2343 ∼ 2322 ∼= 2322 2344 ∼ 2286 ∼= 2286 2345 ∼ 2277 ∼= 2277 2346 ∼ 2295 ∼= 2295 2347 ∼ 2322 ∼= 2322 2348 ∼ 2295 ∼= 2295 2349 ∼ 734 ∼= 730 2350 ∼ 820 ∼= 820 2351 ∼ 820 ∼= 820 2352 ∼ 2352 ∼= 740 2353 ∼ 820 ∼= 820 2354 ∼ 820 ∼= 820 2355 ∼ 2355 ∼= 2355 2356 ∼ 2352 ∼= 740 2357 ∼ 2355 ∼= 2355 2358 ∼ 2358 ∼= 820 2359 ∼ 820 ∼= 820 2360 ∼ 820 ∼= 820 2361 ∼ 2361 ∼= 2361 2362 ∼ 820 ∼= 820 2363 ∼ 820 ∼= 820 2364 ∼ 2364 ∼= 2364 2365 ∼ 2365 ∼= 2365 2366 ∼ 2366 ∼= 2366 2367 ∼ 2367 ∼= 2367 2368 ∼ 2368 ∼= 739 2369 ∼ 2369 ∼= 2369 2370 ∼ 821 ∼= 821 2371 ∼ 2371 ∼= 2371 2372 ∼ 2372 ∼= 2372 2373 ∼ 821 ∼= 821 2374 ∼ 2374 ∼= 821 2375 ∼ 2375 ∼= 2375 2376 ∼ 2376 ∼= 739 2377 ∼ 820 ∼= 820 2378 ∼ 820 ∼= 820 2379 ∼ 2365 ∼= 2365 2380 ∼ 820 ∼= 820 2381 ∼ 820 ∼= 820 2382 ∼ 2366 ∼= 2366 2383 ∼ 2361 ∼= 2361 2384 ∼ 2364 ∼= 2364 2385 ∼ 2367 ∼= 2367 2386 ∼ 820 ∼= 820 2387 ∼ 820 ∼= 820 2388 ∼ 2388 ∼= 821 2389 ∼ 820 ∼= 820 30 Classification of groups generated by automata 2390 ∼ 820 ∼= 820 2391 ∼ 2391 ∼= 2391 2392 ∼ 2388 ∼= 821 2393 ∼ 2391 ∼= 2391 2394 ∼ 2394 ∼= 820 2395 ∼ 2395 ∼= 2395 2396 ∼ 2396 ∼= 2396 2397 ∼ 821 ∼= 821 2398 ∼ 2398 ∼= 2398 2399 ∼ 2399 ∼= 2399 2400 ∼ 821 ∼= 821 2401 ∼ 2401 ∼= 2401 2402 ∼ 2402 ∼= 2402 2403 ∼ 2403 ∼= 2287 2404 ∼ 2368 ∼= 739 2405 ∼ 2371 ∼= 2371 2406 ∼ 2374 ∼= 821 2407 ∼ 2369 ∼= 2369 2408 ∼ 2372 ∼= 2372 2409 ∼ 2375 ∼= 2375 2410 ∼ 821 ∼= 821 2411 ∼ 821 ∼= 821 2412 ∼ 2376 ∼= 739 2413 ∼ 2395 ∼= 2395 2414 ∼ 2398 ∼= 2398 2415 ∼ 2401 ∼= 2401 2416 ∼ 2396 ∼= 2396 2417 ∼ 2399 ∼= 2399 2418 ∼ 2402 ∼= 2402 2419 ∼ 821 ∼= 821 2420 ∼ 821 ∼= 821 2421 ∼ 2403 ∼= 2287 2422 ∼ 2422 ∼= 820 2423 ∼ 2423 ∼= 2423 2424 ∼ 2424 ∼= 966 2425 ∼ 2423 ∼= 2423 2426 ∼ 2426 ∼= 2277 2427 ∼ 2427 ∼= 2427 2428 ∼ 2424 ∼= 966 2429 ∼ 2427 ∼= 2427 2430 ∼ 824 ∼= 820 2431 ∼ 730 ∼= 730 2432 ∼ 730 ∼= 730 2433 ∼ 2271 ∼= 2271 2434 ∼ 730 ∼= 730 2435 ∼ 730 ∼= 730 2436 ∼ 2274 ∼= 2274 2437 ∼ 2271 ∼= 2271 2438 ∼ 2274 ∼= 2274 2439 ∼ 2277 ∼= 2277 2440 ∼ 730 ∼= 730 2441 ∼ 730 ∼= 730 2442 ∼ 2280 ∼= 2280 2443 ∼ 730 ∼= 730 2444 ∼ 730 ∼= 730 2445 ∼ 2283 ∼= 2283 2446 ∼ 2284 ∼= 2284 2447 ∼ 2285 ∼= 2285 2448 ∼ 2286 ∼= 2286 2449 ∼ 2287 ∼= 2287 2450 ∼ 2285 ∼= 2285 2451 ∼ 731 ∼= 731 2452 ∼ 2283 ∼= 2283 2453 ∼ 2274 ∼= 2274 2454 ∼ 731 ∼= 731 2455 ∼ 2293 ∼= 2293 2456 ∼ 2294 ∼= 2294 2457 ∼ 2295 ∼= 2295 2458 ∼ 730 ∼= 730 2459 ∼ 730 ∼= 730 2460 ∼ 2284 ∼= 2284 2461 ∼ 730 ∼= 730 2462 ∼ 730 ∼= 730 2463 ∼ 2285 ∼= 2285 2464 ∼ 2280 ∼= 2280 2465 ∼ 2283 ∼= 2283 2466 ∼ 2286 ∼= 2286 2467 ∼ 730 ∼= 730 2468 ∼ 730 ∼= 730 2469 ∼ 2307 ∼= 2307 2470 ∼ 730 ∼= 730 2471 ∼ 730 ∼= 730 2472 ∼ 2287 ∼= 2287 2473 ∼ 2307 ∼= 2307 2474 ∼ 2287 ∼= 2287 2475 ∼ 2313 ∼= 2277 2476 ∼ 2307 ∼= 2307 2477 ∼ 2284 ∼= 2284 2478 ∼ 731 ∼= 731 2479 ∼ 2280 ∼= 2280 2480 ∼ 2271 ∼= 2271 2481 ∼ 731 ∼= 731 2482 ∼ 2320 ∼= 2294 2483 ∼ 2293 ∼= 2293 2484 ∼ 2322 ∼= 2322 2485 ∼ 2287 ∼= 2287 2486 ∼ 2283 ∼= 2283 2487 ∼ 2293 ∼= 2293 2488 ∼ 2285 ∼= 2285 2489 ∼ 2274 ∼= 2274 2490 ∼ 2294 ∼= 2294 2491 ∼ 731 ∼= 731 2492 ∼ 731 ∼= 731 2493 ∼ 2295 ∼= 2295 2494 ∼ 2307 ∼= 2307 2495 ∼ 2280 ∼= 2280 2496 ∼ 2320 ∼= 2294 2497 ∼ 2284 ∼= 2284 2498 ∼ 2271 ∼= 2271 2499 ∼ 2293 ∼= 2293 2500 ∼ 731 ∼= 731 2501 ∼ 731 ∼= 731 2502 ∼ 2322 ∼= 2322 2503 ∼ 2313 ∼= 2277 2504 ∼ 2286 ∼= 2286 2505 ∼ 2322 ∼= 2322 2506 ∼ 2286 ∼= 2286 2507 ∼ 2277 ∼= 2277 2508 ∼ 2295 ∼= 2295 2509 ∼ 2322 ∼= 2322 2510 ∼ 2295 ∼= 2295 2511 ∼ 734 ∼= 730 2512 ∼ 730 ∼= 730 2513 ∼ 730 ∼= 730 2514 ∼ 2237 ∼= 2237 2515 ∼ 730 ∼= 730 2516 ∼ 730 ∼= 730 2517 ∼ 2210 ∼= 2210 2518 ∼ 2237 ∼= 2237 2519 ∼ 2210 ∼= 2210 2520 ∼ 2264 ∼= 730 2521 ∼ 730 ∼= 730 2522 ∼ 730 ∼= 730 2523 ∼ 2236 ∼= 2236 2524 ∼ 730 ∼= 730 2525 ∼ 730 ∼= 730 2526 ∼ 2209 ∼= 2209 2527 ∼ 2234 ∼= 2234 2528 ∼ 2207 ∼= 2207 2529 ∼ 2261 ∼= 2261 2530 ∼ 2229 ∼= 2229 2531 ∼ 2204 ∼= 2204 2532 ∼ 731 ∼= 731 2533 ∼ 2202 ∼= 2202 2534 ∼ 2193 ∼= 2193 2535 ∼ 731 ∼= 731 2536 ∼ 2240 ∼= 2240 2537 ∼ 2213 ∼= 2213 2538 ∼ 2265 ∼= 2265 2539 ∼ 730 ∼= 730 2540 ∼ 730 ∼= 730 2541 ∼ 2234 ∼= 2234 2542 ∼ 730 ∼= 730 2543 ∼ 730 ∼= 730 2544 ∼ 2207 ∼= 2207 2545 ∼ 2236 ∼= 2236 2546 ∼ 2209 ∼= 2209 2547 ∼ 2261 ∼= 2261 2548 ∼ 730 ∼= 730 2549 ∼ 730 ∼= 730 2550 ∼ 2233 ∼= 2233 2551 ∼ 730 ∼= 730 2552 ∼ 730 ∼= 730 2553 ∼ 2206 ∼= 748 2554 ∼ 2233 ∼= 2233 2555 ∼ 2206 ∼= 748 2556 ∼ 2260 ∼= 802 2557 ∼ 2226 ∼= 820 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 2558 ∼ 2203 ∼= 2203 2559 ∼ 731 ∼= 731 2560 ∼ 2199 ∼= 2199 2561 ∼ 2190 ∼= 750 2562 ∼ 731 ∼= 731 2563 ∼ 2239 ∼= 2239 2564 ∼ 2212 ∼= 2212 2565 ∼ 2262 ∼= 750 2566 ∼ 2229 ∼= 2229 2567 ∼ 2202 ∼= 2202 2568 ∼ 2240 ∼= 2240 2569 ∼ 2204 ∼= 2204 2570 ∼ 2193 ∼= 2193 2571 ∼ 2213 ∼= 2213 2572 ∼ 731 ∼= 731 2573 ∼ 731 ∼= 731 2574 ∼ 2265 ∼= 2265 2575 ∼ 2226 ∼= 820 2576 ∼ 2199 ∼= 2199 2577 ∼ 2239 ∼= 2239 2578 ∼ 2203 ∼= 2203 2579 ∼ 2190 ∼= 750 2580 ∼ 2212 ∼= 2212 2581 ∼ 731 ∼= 731 2582 ∼ 731 ∼= 731 2583 ∼ 2262 ∼= 750 2584 ∼ 2232 ∼= 730 2585 ∼ 2205 ∼= 775 2586 ∼ 2241 ∼= 739 2587 ∼ 2205 ∼= 775 2588 ∼ 2196 ∼= 802 2589 ∼ 2214 ∼= 748 2590 ∼ 2241 ∼= 739 2591 ∼ 2214 ∼= 748 2592 ∼ 734 ∼= 730 2593 ∼ 820 ∼= 820 2594 ∼ 820 ∼= 820 2595 ∼ 2399 ∼= 2399 2596 ∼ 820 ∼= 820 2597 ∼ 820 ∼= 820 2598 ∼ 2372 ∼= 2372 2599 ∼ 2399 ∼= 2399 2600 ∼ 2372 ∼= 2372 2601 ∼ 2426 ∼= 2277 2602 ∼ 820 ∼= 820 2603 ∼ 820 ∼= 820 2604 ∼ 2398 ∼= 2398 2605 ∼ 820 ∼= 820 2606 ∼ 820 ∼= 820 2607 ∼ 2371 ∼= 2371 2608 ∼ 2396 ∼= 2396 2609 ∼ 2369 ∼= 2369 2610 ∼ 2423 ∼= 2423 2611 ∼ 2391 ∼= 2391 2612 ∼ 2366 ∼= 2366 2613 ∼ 821 ∼= 821 2614 ∼ 2364 ∼= 2364 2615 ∼ 2355 ∼= 2355 2616 ∼ 821 ∼= 821 2617 ∼ 2402 ∼= 2402 2618 ∼ 2375 ∼= 2375 2619 ∼ 2427 ∼= 2427 2620 ∼ 820 ∼= 820 2621 ∼ 820 ∼= 820 2622 ∼ 2396 ∼= 2396 2623 ∼ 820 ∼= 820 2624 ∼ 820 ∼= 820 2625 ∼ 2369 ∼= 2369 2626 ∼ 2398 ∼= 2398 2627 ∼ 2371 ∼= 2371 2628 ∼ 2423 ∼= 2423 2629 ∼ 820 ∼= 820 2630 ∼ 820 ∼= 820 2631 ∼ 2395 ∼= 2395 2632 ∼ 820 ∼= 820 2633 ∼ 820 ∼= 820 2634 ∼ 2368 ∼= 739 2635 ∼ 2395 ∼= 2395 2636 ∼ 2368 ∼= 739 2637 ∼ 2422 ∼= 820 2638 ∼ 2388 ∼= 821 2639 ∼ 2365 ∼= 2365 2640 ∼ 821 ∼= 821 2641 ∼ 2361 ∼= 2361 2642 ∼ 2352 ∼= 740 2643 ∼ 821 ∼= 821 2644 ∼ 2401 ∼= 2401 2645 ∼ 2374 ∼= 821 2646 ∼ 2424 ∼= 966 2647 ∼ 2391 ∼= 2391 2648 ∼ 2364 ∼= 2364 2649 ∼ 2402 ∼= 2402 2650 ∼ 2366 ∼= 2366 2651 ∼ 2355 ∼= 2355 2652 ∼ 2375 ∼= 2375 2653 ∼ 821 ∼= 821 2654 ∼ 821 ∼= 821 2655 ∼ 2427 ∼= 2427 2656 ∼ 2388 ∼= 821 2657 ∼ 2361 ∼= 2361 2658 ∼ 2401 ∼= 2401 2659 ∼ 2365 ∼= 2365 2660 ∼ 2352 ∼= 740 2661 ∼ 2374 ∼= 821 2662 ∼ 821 ∼= 821 2663 ∼ 821 ∼= 821 2664 ∼ 2424 ∼= 966 2665 ∼ 2394 ∼= 820 2666 ∼ 2367 ∼= 2367 2667 ∼ 2403 ∼= 2287 2668 ∼ 2367 ∼= 2367 2669 ∼ 2358 ∼= 820 2670 ∼ 2376 ∼= 739 2671 ∼ 2403 ∼= 2287 2672 ∼ 2376 ∼= 739 2673 ∼ 824 ∼= 820 2674 ∼ 820 ∼= 820 2675 ∼ 820 ∼= 820 2676 ∼ 2352 ∼= 740 2677 ∼ 820 ∼= 820 2678 ∼ 820 ∼= 820 2679 ∼ 2355 ∼= 2355 2680 ∼ 2352 ∼= 740 2681 ∼ 2355 ∼= 2355 2682 ∼ 2358 ∼= 820 2683 ∼ 820 ∼= 820 2684 ∼ 820 ∼= 820 2685 ∼ 2361 ∼= 2361 2686 ∼ 820 ∼= 820 2687 ∼ 820 ∼= 820 2688 ∼ 2364 ∼= 2364 2689 ∼ 2365 ∼= 2365 2690 ∼ 2366 ∼= 2366 2691 ∼ 2367 ∼= 2367 2692 ∼ 2368 ∼= 739 2693 ∼ 2369 ∼= 2369 2694 ∼ 821 ∼= 821 2695 ∼ 2371 ∼= 2371 2696 ∼ 2372 ∼= 2372 2697 ∼ 821 ∼= 821 2698 ∼ 2374 ∼= 821 2699 ∼ 2375 ∼= 2375 2700 ∼ 2376 ∼= 739 2701 ∼ 820 ∼= 820 2702 ∼ 820 ∼= 820 2703 ∼ 2365 ∼= 2365 2704 ∼ 820 ∼= 820 2705 ∼ 820 ∼= 820 2706 ∼ 2366 ∼= 2366 2707 ∼ 2361 ∼= 2361 2708 ∼ 2364 ∼= 2364 2709 ∼ 2367 ∼= 2367 2710 ∼ 820 ∼= 820 2711 ∼ 820 ∼= 820 2712 ∼ 2388 ∼= 821 2713 ∼ 820 ∼= 820 2714 ∼ 820 ∼= 820 2715 ∼ 2391 ∼= 2391 2716 ∼ 2388 ∼= 821 2717 ∼ 2391 ∼= 2391 2718 ∼ 2394 ∼= 820 2719 ∼ 2395 ∼= 2395 2720 ∼ 2396 ∼= 2396 2721 ∼ 821 ∼= 821 2722 ∼ 2398 ∼= 2398 2723 ∼ 2399 ∼= 2399 2724 ∼ 821 ∼= 821 2725 ∼ 2401 ∼= 2401 32 Classification of groups generated by automata 2726 ∼ 2402 ∼= 2402 2727 ∼ 2403 ∼= 2287 2728 ∼ 2368 ∼= 739 2729 ∼ 2371 ∼= 2371 2730 ∼ 2374 ∼= 821 2731 ∼ 2369 ∼= 2369 2732 ∼ 2372 ∼= 2372 2733 ∼ 2375 ∼= 2375 2734 ∼ 821 ∼= 821 2735 ∼ 821 ∼= 821 2736 ∼ 2376 ∼= 739 2737 ∼ 2395 ∼= 2395 2738 ∼ 2398 ∼= 2398 2739 ∼ 2401 ∼= 2401 2740 ∼ 2396 ∼= 2396 2741 ∼ 2399 ∼= 2399 2742 ∼ 2402 ∼= 2402 2743 ∼ 821 ∼= 821 2744 ∼ 821 ∼= 821 2745 ∼ 2403 ∼= 2287 2746 ∼ 2422 ∼= 820 2747 ∼ 2423 ∼= 2423 2748 ∼ 2424 ∼= 966 2749 ∼ 2423 ∼= 2423 2750 ∼ 2426 ∼= 2277 2751 ∼ 2427 ∼= 2427 2752 ∼ 2424 ∼= 966 2753 ∼ 2427 ∼= 2427 2754 ∼ 824 ∼= 820 2755 ∼ 820 ∼= 820 2756 ∼ 820 ∼= 820 2757 ∼ 2399 ∼= 2399 2758 ∼ 820 ∼= 820 2759 ∼ 820 ∼= 820 2760 ∼ 2372 ∼= 2372 2761 ∼ 2399 ∼= 2399 2762 ∼ 2372 ∼= 2372 2763 ∼ 2426 ∼= 2277 2764 ∼ 820 ∼= 820 2765 ∼ 820 ∼= 820 2766 ∼ 2398 ∼= 2398 2767 ∼ 820 ∼= 820 2768 ∼ 820 ∼= 820 2769 ∼ 2371 ∼= 2371 2770 ∼ 2396 ∼= 2396 2771 ∼ 2369 ∼= 2369 2772 ∼ 2423 ∼= 2423 2773 ∼ 2391 ∼= 2391 2774 ∼ 2366 ∼= 2366 2775 ∼ 821 ∼= 821 2776 ∼ 2364 ∼= 2364 2777 ∼ 2355 ∼= 2355 2778 ∼ 821 ∼= 821 2779 ∼ 2402 ∼= 2402 2780 ∼ 2375 ∼= 2375 2781 ∼ 2427 ∼= 2427 2782 ∼ 820 ∼= 820 2783 ∼ 820 ∼= 820 2784 ∼ 2396 ∼= 2396 2785 ∼ 820 ∼= 820 2786 ∼ 820 ∼= 820 2787 ∼ 2369 ∼= 2369 2788 ∼ 2398 ∼= 2398 2789 ∼ 2371 ∼= 2371 2790 ∼ 2423 ∼= 2423 2791 ∼ 820 ∼= 820 2792 ∼ 820 ∼= 820 2793 ∼ 2395 ∼= 2395 2794 ∼ 820 ∼= 820 2795 ∼ 820 ∼= 820 2796 ∼ 2368 ∼= 739 2797 ∼ 2395 ∼= 2395 2798 ∼ 2368 ∼= 739 2799 ∼ 2422 ∼= 820 2800 ∼ 2388 ∼= 821 2801 ∼ 2365 ∼= 2365 2802 ∼ 821 ∼= 821 2803 ∼ 2361 ∼= 2361 2804 ∼ 2352 ∼= 740 2805 ∼ 821 ∼= 821 2806 ∼ 2401 ∼= 2401 2807 ∼ 2374 ∼= 821 2808 ∼ 2424 ∼= 966 2809 ∼ 2391 ∼= 2391 2810 ∼ 2364 ∼= 2364 2811 ∼ 2402 ∼= 2402 2812 ∼ 2366 ∼= 2366 2813 ∼ 2355 ∼= 2355 2814 ∼ 2375 ∼= 2375 2815 ∼ 821 ∼= 821 2816 ∼ 821 ∼= 821 2817 ∼ 2427 ∼= 2427 2818 ∼ 2388 ∼= 821 2819 ∼ 2361 ∼= 2361 2820 ∼ 2401 ∼= 2401 2821 ∼ 2365 ∼= 2365 2822 ∼ 2352 ∼= 740 2823 ∼ 2374 ∼= 821 2824 ∼ 821 ∼= 821 2825 ∼ 821 ∼= 821 2826 ∼ 2424 ∼= 966 2827 ∼ 2394 ∼= 820 2828 ∼ 2367 ∼= 2367 2829 ∼ 2403 ∼= 2287 2830 ∼ 2367 ∼= 2367 2831 ∼ 2358 ∼= 820 2832 ∼ 2376 ∼= 739 2833 ∼ 2403 ∼= 2287 2834 ∼ 2376 ∼= 739 2835 ∼ 824 ∼= 820 2836 ∼ 1090 ∼= 1090 2837 ∼ 1090 ∼= 1090 2838 ∼ 2838 ∼= 750 2839 ∼ 1090 ∼= 1090 2840 ∼ 1090 ∼= 1090 2841 ∼ 2841 ∼= 2841 2842 ∼ 2838 ∼= 750 2843 ∼ 2841 ∼= 2841 2844 ∼ 2844 ∼= 730 2845 ∼ 1090 ∼= 1090 2846 ∼ 1090 ∼= 1090 2847 ∼ 2847 ∼= 929 2848 ∼ 1090 ∼= 1090 2849 ∼ 1090 ∼= 1090 2850 ∼ 2850 ∼= 2850 2851 ∼ 2851 ∼= 929 2852 ∼ 2852 ∼= 849 2853 ∼ 2853 ∼= 2853 2854 ∼ 2854 ∼= 847 2855 ∼ 2852 ∼= 849 2856 ∼ 1091 ∼= 731 2857 ∼ 2850 ∼= 2850 2858 ∼ 2841 ∼= 2841 2859 ∼ 1091 ∼= 731 2860 ∼ 2860 ∼= 2212 2861 ∼ 2861 ∼= 731 2862 ∼ 2862 ∼= 847 2863 ∼ 1090 ∼= 1090 2864 ∼ 1090 ∼= 1090 2865 ∼ 2851 ∼= 929 2866 ∼ 1090 ∼= 1090 2867 ∼ 1090 ∼= 1090 2868 ∼ 2852 ∼= 849 2869 ∼ 2847 ∼= 929 2870 ∼ 2850 ∼= 2850 2871 ∼ 2853 ∼= 2853 2872 ∼ 1090 ∼= 1090 2873 ∼ 1090 ∼= 1090 2874 ∼ 2874 ∼= 820 2875 ∼ 1090 ∼= 1090 2876 ∼ 1090 ∼= 1090 2877 ∼ 2854 ∼= 847 2878 ∼ 2874 ∼= 820 2879 ∼ 2854 ∼= 847 2880 ∼ 2880 ∼= 730 2881 ∼ 2874 ∼= 820 2882 ∼ 2851 ∼= 929 2883 ∼ 1091 ∼= 731 2884 ∼ 2847 ∼= 929 2885 ∼ 2838 ∼= 750 2886 ∼ 1091 ∼= 731 2887 ∼ 2887 ∼= 731 2888 ∼ 2860 ∼= 2212 2889 ∼ 2889 ∼= 750 2890 ∼ 2854 ∼= 847 2891 ∼ 2850 ∼= 2850 2892 ∼ 2860 ∼= 2212 2893 ∼ 2852 ∼= 849 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 2894 ∼ 2841 ∼= 2841 2895 ∼ 2861 ∼= 731 2896 ∼ 1091 ∼= 731 2897 ∼ 1091 ∼= 731 2898 ∼ 2862 ∼= 847 2899 ∼ 2874 ∼= 820 2900 ∼ 2847 ∼= 929 2901 ∼ 2887 ∼= 731 2902 ∼ 2851 ∼= 929 2903 ∼ 2838 ∼= 750 2904 ∼ 2860 ∼= 2212 2905 ∼ 1091 ∼= 731 2906 ∼ 1091 ∼= 731 2907 ∼ 2889 ∼= 750 2908 ∼ 2880 ∼= 730 2909 ∼ 2853 ∼= 2853 2910 ∼ 2889 ∼= 750 2911 ∼ 2853 ∼= 2853 2912 ∼ 2844 ∼= 730 2913 ∼ 2862 ∼= 847 2914 ∼ 2889 ∼= 750 2915 ∼ 2862 ∼= 847 2916 ∼ 1094 ∼= 1090 2917 ∼ 1094 ∼= 1090 2918 ∼ 1094 ∼= 1090 2919 ∼ 972 ∼= 739 2920 ∼ 1094 ∼= 1090 2921 ∼ 1094 ∼= 1090 2922 ∼ 891 ∼= 891 2923 ∼ 972 ∼= 739 2924 ∼ 891 ∼= 891 2925 ∼ 810 ∼= 802 2926 ∼ 1094 ∼= 1090 2927 ∼ 1094 ∼= 1090 2928 ∼ 945 ∼= 941 2929 ∼ 1094 ∼= 1090 2930 ∼ 1094 ∼= 1090 2931 ∼ 864 ∼= 864 2932 ∼ 963 ∼= 963 2933 ∼ 882 ∼= 882 2934 ∼ 783 ∼= 775 2935 ∼ 851 ∼= 847 2936 ∼ 878 ∼= 878 2937 ∼ 824 ∼= 820 2938 ∼ 860 ∼= 860 2939 ∼ 887 ∼= 887 2940 ∼ 824 ∼= 820 2941 ∼ 842 ∼= 838 2942 ∼ 869 ∼= 869 2943 ∼ 756 ∼= 748 2944 ∼ 1094 ∼= 1090 2945 ∼ 1094 ∼= 1090 2946 ∼ 963 ∼= 963 2947 ∼ 1094 ∼= 1090 2948 ∼ 1094 ∼= 1090 2949 ∼ 882 ∼= 882 2950 ∼ 945 ∼= 941 2951 ∼ 864 ∼= 864 2952 ∼ 783 ∼= 775 2953 ∼ 1094 ∼= 1090 2954 ∼ 1094 ∼= 1090 2955 ∼ 936 ∼= 820 2956 ∼ 1094 ∼= 1090 2957 ∼ 1094 ∼= 1090 2958 ∼ 855 ∼= 847 2959 ∼ 936 ∼= 820 2960 ∼ 855 ∼= 847 2961 ∼ 774 ∼= 730 2962 ∼ 932 ∼= 820 2963 ∼ 959 ∼= 959 2964 ∼ 824 ∼= 820 2965 ∼ 941 ∼= 941 2966 ∼ 968 ∼= 968 2967 ∼ 824 ∼= 820 2968 ∼ 923 ∼= 923 2969 ∼ 846 ∼= 846 2970 ∼ 747 ∼= 739 2971 ∼ 851 ∼= 847 2972 ∼ 860 ∼= 860 2973 ∼ 842 ∼= 838 2974 ∼ 878 ∼= 878 2975 ∼ 887 ∼= 887 2976 ∼ 869 ∼= 869 2977 ∼ 824 ∼= 820 2978 ∼ 824 ∼= 820 2979 ∼ 756 ∼= 748 2980 ∼ 932 ∼= 820 2981 ∼ 941 ∼= 941 2982 ∼ 923 ∼= 923 2983 ∼ 959 ∼= 959 2984 ∼ 968 ∼= 968 2985 ∼ 846 ∼= 846 2986 ∼ 824 ∼= 820 2987 ∼ 824 ∼= 820 2988 ∼ 747 ∼= 739 2989 ∼ 770 ∼= 730 2990 ∼ 779 ∼= 779 2991 ∼ 743 ∼= 739 2992 ∼ 779 ∼= 779 2993 ∼ 806 ∼= 802 2994 ∼ 752 ∼= 752 2995 ∼ 743 ∼= 739 2996 ∼ 752 ∼= 752 2997 ∼ 734 ∼= 730 2998 ∼ 1094 ∼= 1090 2999 ∼ 1094 ∼= 1090 3000 ∼ 969 ∼= 969 3001 ∼ 1094 ∼= 1090 3002 ∼ 1094 ∼= 1090 3003 ∼ 888 ∼= 888 3004 ∼ 969 ∼= 969 3005 ∼ 888 ∼= 888 3006 ∼ 807 ∼= 771 3007 ∼ 1094 ∼= 1090 3008 ∼ 1094 ∼= 1090 3009 ∼ 942 ∼= 942 3010 ∼ 1094 ∼= 1090 3011 ∼ 1094 ∼= 1090 3012 ∼ 861 ∼= 861 3013 ∼ 960 ∼= 960 3014 ∼ 879 ∼= 879 3015 ∼ 780 ∼= 780 3016 ∼ 852 ∼= 852 3017 ∼ 879 ∼= 879 3018 ∼ 824 ∼= 820 3019 ∼ 861 ∼= 861 3020 ∼ 888 ∼= 888 3021 ∼ 824 ∼= 820 3022 ∼ 843 ∼= 843 3023 ∼ 870 ∼= 870 3024 ∼ 753 ∼= 753 3025 ∼ 1094 ∼= 1090 3026 ∼ 1094 ∼= 1090 3027 ∼ 960 ∼= 960 3028 ∼ 1094 ∼= 1090 3029 ∼ 1094 ∼= 1090 3030 ∼ 879 ∼= 879 3031 ∼ 942 ∼= 942 3032 ∼ 861 ∼= 861 3033 ∼ 780 ∼= 780 3034 ∼ 1094 ∼= 1090 3035 ∼ 1094 ∼= 1090 3036 ∼ 933 ∼= 849 3037 ∼ 1094 ∼= 1090 3038 ∼ 1094 ∼= 1090 3039 ∼ 852 ∼= 852 3040 ∼ 933 ∼= 849 3041 ∼ 852 ∼= 852 3042 ∼ 771 ∼= 771 3043 ∼ 933 ∼= 849 3044 ∼ 960 ∼= 960 3045 ∼ 824 ∼= 820 3046 ∼ 942 ∼= 942 3047 ∼ 969 ∼= 969 3048 ∼ 824 ∼= 820 3049 ∼ 924 ∼= 870 3050 ∼ 843 ∼= 843 3051 ∼ 744 ∼= 744 3052 ∼ 852 ∼= 852 3053 ∼ 861 ∼= 861 3054 ∼ 843 ∼= 843 3055 ∼ 879 ∼= 879 3056 ∼ 888 ∼= 888 3057 ∼ 870 ∼= 870 3058 ∼ 824 ∼= 820 3059 ∼ 824 ∼= 820 3060 ∼ 753 ∼= 753 3061 ∼ 933 ∼= 849 34 Classification of groups generated by automata 3062 ∼ 942 ∼= 942 3063 ∼ 924 ∼= 870 3064 ∼ 960 ∼= 960 3065 ∼ 969 ∼= 969 3066 ∼ 843 ∼= 843 3067 ∼ 824 ∼= 820 3068 ∼ 824 ∼= 820 3069 ∼ 744 ∼= 744 3070 ∼ 771 ∼= 771 3071 ∼ 780 ∼= 780 3072 ∼ 744 ∼= 744 3073 ∼ 780 ∼= 780 3074 ∼ 807 ∼= 771 3075 ∼ 753 ∼= 753 3076 ∼ 744 ∼= 744 3077 ∼ 753 ∼= 753 3078 ∼ 734 ∼= 730 3079 ∼ 1091 ∼= 731 3080 ∼ 1091 ∼= 731 3081 ∼ 966 ∼= 966 3082 ∼ 1091 ∼= 731 3083 ∼ 1091 ∼= 731 3084 ∼ 885 ∼= 885 3085 ∼ 966 ∼= 966 3086 ∼ 885 ∼= 885 3087 ∼ 804 ∼= 731 3088 ∼ 1091 ∼= 731 3089 ∼ 1091 ∼= 731 3090 ∼ 939 ∼= 939 3091 ∼ 1091 ∼= 731 3092 ∼ 1091 ∼= 731 3093 ∼ 858 ∼= 858 3094 ∼ 957 ∼= 957 3095 ∼ 876 ∼= 876 3096 ∼ 777 ∼= 777 3097 ∼ 848 ∼= 750 3098 ∼ 875 ∼= 875 3099 ∼ 821 ∼= 821 3100 ∼ 857 ∼= 857 3101 ∼ 884 ∼= 884 3102 ∼ 821 ∼= 821 3103 ∼ 839 ∼= 821 3104 ∼ 866 ∼= 866 3105 ∼ 750 ∼= 750 3106 ∼ 1091 ∼= 731 3107 ∼ 1091 ∼= 731 3108 ∼ 957 ∼= 957 3109 ∼ 1091 ∼= 731 3110 ∼ 1091 ∼= 731 3111 ∼ 876 ∼= 876 3112 ∼ 939 ∼= 939 3113 ∼ 858 ∼= 858 3114 ∼ 777 ∼= 777 3115 ∼ 1091 ∼= 731 3116 ∼ 1091 ∼= 731 3117 ∼ 930 ∼= 821 3118 ∼ 1091 ∼= 731 3119 ∼ 1091 ∼= 731 3120 ∼ 849 ∼= 849 3121 ∼ 930 ∼= 821 3122 ∼ 849 ∼= 849 3123 ∼ 768 ∼= 731 3124 ∼ 929 ∼= 929 3125 ∼ 956 ∼= 956 3126 ∼ 821 ∼= 821 3127 ∼ 938 ∼= 938 3128 ∼ 965 ∼= 965 3129 ∼ 821 ∼= 821 3130 ∼ 920 ∼= 920 3131 ∼ 840 ∼= 840 3132 ∼ 741 ∼= 741 3133 ∼ 848 ∼= 750 3134 ∼ 857 ∼= 857 3135 ∼ 839 ∼= 821 3136 ∼ 875 ∼= 875 3137 ∼ 884 ∼= 884 3138 ∼ 866 ∼= 866 3139 ∼ 821 ∼= 821 3140 ∼ 821 ∼= 821 3141 ∼ 750 ∼= 750 3142 ∼ 929 ∼= 929 3143 ∼ 938 ∼= 938 3144 ∼ 920 ∼= 920 3145 ∼ 956 ∼= 956 3146 ∼ 965 ∼= 965 3147 ∼ 840 ∼= 840 3148 ∼ 821 ∼= 821 3149 ∼ 821 ∼= 821 3150 ∼ 741 ∼= 741 3151 ∼ 767 ∼= 731 3152 ∼ 776 ∼= 776 3153 ∼ 740 ∼= 740 3154 ∼ 776 ∼= 776 3155 ∼ 803 ∼= 771 3156 ∼ 749 ∼= 749 3157 ∼ 740 ∼= 740 3158 ∼ 749 ∼= 749 3159 ∼ 731 ∼= 731 3160 ∼ 1094 ∼= 1090 3161 ∼ 1094 ∼= 1090 3162 ∼ 969 ∼= 969 3163 ∼ 1094 ∼= 1090 3164 ∼ 1094 ∼= 1090 3165 ∼ 888 ∼= 888 3166 ∼ 969 ∼= 969 3167 ∼ 888 ∼= 888 3168 ∼ 807 ∼= 771 3169 ∼ 1094 ∼= 1090 3170 ∼ 1094 ∼= 1090 3171 ∼ 942 ∼= 942 3172 ∼ 1094 ∼= 1090 3173 ∼ 1094 ∼= 1090 3174 ∼ 861 ∼= 861 3175 ∼ 960 ∼= 960 3176 ∼ 879 ∼= 879 3177 ∼ 780 ∼= 780 3178 ∼ 852 ∼= 852 3179 ∼ 879 ∼= 879 3180 ∼ 824 ∼= 820 3181 ∼ 861 ∼= 861 3182 ∼ 888 ∼= 888 3183 ∼ 824 ∼= 820 3184 ∼ 843 ∼= 843 3185 ∼ 870 ∼= 870 3186 ∼ 753 ∼= 753 3187 ∼ 1094 ∼= 1090 3188 ∼ 1094 ∼= 1090 3189 ∼ 960 ∼= 960 3190 ∼ 1094 ∼= 1090 3191 ∼ 1094 ∼= 1090 3192 ∼ 879 ∼= 879 3193 ∼ 942 ∼= 942 3194 ∼ 861 ∼= 861 3195 ∼ 780 ∼= 780 3196 ∼ 1094 ∼= 1090 3197 ∼ 1094 ∼= 1090 3198 ∼ 933 ∼= 849 3199 ∼ 1094 ∼= 1090 3200 ∼ 1094 ∼= 1090 3201 ∼ 852 ∼= 852 3202 ∼ 933 ∼= 849 3203 ∼ 852 ∼= 852 3204 ∼ 771 ∼= 771 3205 ∼ 933 ∼= 849 3206 ∼ 960 ∼= 960 3207 ∼ 824 ∼= 820 3208 ∼ 942 ∼= 942 3209 ∼ 969 ∼= 969 3210 ∼ 824 ∼= 820 3211 ∼ 924 ∼= 870 3212 ∼ 843 ∼= 843 3213 ∼ 744 ∼= 744 3214 ∼ 852 ∼= 852 3215 ∼ 861 ∼= 861 3216 ∼ 843 ∼= 843 3217 ∼ 879 ∼= 879 3218 ∼ 888 ∼= 888 3219 ∼ 870 ∼= 870 3220 ∼ 824 ∼= 820 3221 ∼ 824 ∼= 820 3222 ∼ 753 ∼= 753 3223 ∼ 933 ∼= 849 3224 ∼ 942 ∼= 942 3225 ∼ 924 ∼= 870 3226 ∼ 960 ∼= 960 3227 ∼ 969 ∼= 969 3228 ∼ 843 ∼= 843 3229 ∼ 824 ∼= 820 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 3230 ∼ 824 ∼= 820 3231 ∼ 744 ∼= 744 3232 ∼ 771 ∼= 771 3233 ∼ 780 ∼= 780 3234 ∼ 744 ∼= 744 3235 ∼ 780 ∼= 780 3236 ∼ 807 ∼= 771 3237 ∼ 753 ∼= 753 3238 ∼ 744 ∼= 744 3239 ∼ 753 ∼= 753 3240 ∼ 734 ∼= 730 3241 ∼ 1094 ∼= 1090 3242 ∼ 1094 ∼= 1090 3243 ∼ 968 ∼= 968 3244 ∼ 1094 ∼= 1090 3245 ∼ 1094 ∼= 1090 3246 ∼ 887 ∼= 887 3247 ∼ 968 ∼= 968 3248 ∼ 887 ∼= 887 3249 ∼ 806 ∼= 802 3250 ∼ 1094 ∼= 1090 3251 ∼ 1094 ∼= 1090 3252 ∼ 941 ∼= 941 3253 ∼ 1094 ∼= 1090 3254 ∼ 1094 ∼= 1090 3255 ∼ 860 ∼= 860 3256 ∼ 959 ∼= 959 3257 ∼ 878 ∼= 878 3258 ∼ 779 ∼= 779 3259 ∼ 855 ∼= 847 3260 ∼ 882 ∼= 882 3261 ∼ 824 ∼= 820 3262 ∼ 864 ∼= 864 3263 ∼ 891 ∼= 891 3264 ∼ 824 ∼= 820 3265 ∼ 846 ∼= 846 3266 ∼ 869 ∼= 869 3267 ∼ 752 ∼= 752 3268 ∼ 1094 ∼= 1090 3269 ∼ 1094 ∼= 1090 3270 ∼ 959 ∼= 959 3271 ∼ 1094 ∼= 1090 3272 ∼ 1094 ∼= 1090 3273 ∼ 878 ∼= 878 3274 ∼ 941 ∼= 941 3275 ∼ 860 ∼= 860 3276 ∼ 779 ∼= 779 3277 ∼ 1094 ∼= 1090 3278 ∼ 1094 ∼= 1090 3279 ∼ 932 ∼= 820 3280 ∼ 1094 ∼= 1090 3281 ∼ 1094 ∼= 1090 3282 ∼ 851 ∼= 847 3283 ∼ 932 ∼= 820 3284 ∼ 851 ∼= 847 3285 ∼ 770 ∼= 730 3286 ∼ 936 ∼= 820 3287 ∼ 963 ∼= 963 3288 ∼ 824 ∼= 820 3289 ∼ 945 ∼= 941 3290 ∼ 972 ∼= 739 3291 ∼ 824 ∼= 820 3292 ∼ 923 ∼= 923 3293 ∼ 842 ∼= 838 3294 ∼ 743 ∼= 739 3295 ∼ 855 ∼= 847 3296 ∼ 864 ∼= 864 3297 ∼ 846 ∼= 846 3298 ∼ 882 ∼= 882 3299 ∼ 891 ∼= 891 3300 ∼ 869 ∼= 869 3301 ∼ 824 ∼= 820 3302 ∼ 824 ∼= 820 3303 ∼ 752 ∼= 752 3304 ∼ 936 ∼= 820 3305 ∼ 945 ∼= 941 3306 ∼ 923 ∼= 923 3307 ∼ 963 ∼= 963 3308 ∼ 972 ∼= 739 3309 ∼ 842 ∼= 838 3310 ∼ 824 ∼= 820 3311 ∼ 824 ∼= 820 3312 ∼ 743 ∼= 739 3313 ∼ 774 ∼= 730 3314 ∼ 783 ∼= 775 3315 ∼ 747 ∼= 739 3316 ∼ 783 ∼= 775 3317 ∼ 810 ∼= 802 3318 ∼ 756 ∼= 748 3319 ∼ 747 ∼= 739 3320 ∼ 756 ∼= 748 3321 ∼ 734 ∼= 730 3322 ∼ 1091 ∼= 731 3323 ∼ 1091 ∼= 731 3324 ∼ 965 ∼= 965 3325 ∼ 1091 ∼= 731 3326 ∼ 1091 ∼= 731 3327 ∼ 884 ∼= 884 3328 ∼ 965 ∼= 965 3329 ∼ 884 ∼= 884 3330 ∼ 803 ∼= 771 3331 ∼ 1091 ∼= 731 3332 ∼ 1091 ∼= 731 3333 ∼ 938 ∼= 938 3334 ∼ 1091 ∼= 731 3335 ∼ 1091 ∼= 731 3336 ∼ 857 ∼= 857 3337 ∼ 956 ∼= 956 3338 ∼ 875 ∼= 875 3339 ∼ 776 ∼= 776 3340 ∼ 849 ∼= 849 3341 ∼ 876 ∼= 876 3342 ∼ 821 ∼= 821 3343 ∼ 858 ∼= 858 3344 ∼ 885 ∼= 885 3345 ∼ 821 ∼= 821 3346 ∼ 840 ∼= 840 3347 ∼ 866 ∼= 866 3348 ∼ 749 ∼= 749 3349 ∼ 1091 ∼= 731 3350 ∼ 1091 ∼= 731 3351 ∼ 956 ∼= 956 3352 ∼ 1091 ∼= 731 3353 ∼ 1091 ∼= 731 3354 ∼ 875 ∼= 875 3355 ∼ 938 ∼= 938 3356 ∼ 857 ∼= 857 3357 ∼ 776 ∼= 776 3358 ∼ 1091 ∼= 731 3359 ∼ 1091 ∼= 731 3360 ∼ 929 ∼= 929 3361 ∼ 1091 ∼= 731 3362 ∼ 1091 ∼= 731 3363 ∼ 848 ∼= 750 3364 ∼ 929 ∼= 929 3365 ∼ 848 ∼= 750 3366 ∼ 767 ∼= 731 3367 ∼ 930 ∼= 821 3368 ∼ 957 ∼= 957 3369 ∼ 821 ∼= 821 3370 ∼ 939 ∼= 939 3371 ∼ 966 ∼= 966 3372 ∼ 821 ∼= 821 3373 ∼ 920 ∼= 920 3374 ∼ 839 ∼= 821 3375 ∼ 740 ∼= 740 3376 ∼ 849 ∼= 849 3377 ∼ 858 ∼= 858 3378 ∼ 840 ∼= 840 3379 ∼ 876 ∼= 876 3380 ∼ 885 ∼= 885 3381 ∼ 866 ∼= 866 3382 ∼ 821 ∼= 821 3383 ∼ 821 ∼= 821 3384 ∼ 749 ∼= 749 3385 ∼ 930 ∼= 821 3386 ∼ 939 ∼= 939 3387 ∼ 920 ∼= 920 3388 ∼ 957 ∼= 957 3389 ∼ 966 ∼= 966 3390 ∼ 839 ∼= 821 3391 ∼ 821 ∼= 821 3392 ∼ 821 ∼= 821 3393 ∼ 740 ∼= 740 3394 ∼ 768 ∼= 731 3395 ∼ 777 ∼= 777 3396 ∼ 741 ∼= 741 3397 ∼ 777 ∼= 777 36 Classification of groups generated by automata 3398 ∼ 804 ∼= 731 3399 ∼ 750 ∼= 750 3400 ∼ 741 ∼= 741 3401 ∼ 750 ∼= 750 3402 ∼ 731 ∼= 731 3403 ∼ 1091 ∼= 731 3404 ∼ 1091 ∼= 731 3405 ∼ 966 ∼= 966 3406 ∼ 1091 ∼= 731 3407 ∼ 1091 ∼= 731 3408 ∼ 885 ∼= 885 3409 ∼ 966 ∼= 966 3410 ∼ 885 ∼= 885 3411 ∼ 804 ∼= 731 3412 ∼ 1091 ∼= 731 3413 ∼ 1091 ∼= 731 3414 ∼ 939 ∼= 939 3415 ∼ 1091 ∼= 731 3416 ∼ 1091 ∼= 731 3417 ∼ 858 ∼= 858 3418 ∼ 957 ∼= 957 3419 ∼ 876 ∼= 876 3420 ∼ 777 ∼= 777 3421 ∼ 848 ∼= 750 3422 ∼ 875 ∼= 875 3423 ∼ 821 ∼= 821 3424 ∼ 857 ∼= 857 3425 ∼ 884 ∼= 884 3426 ∼ 821 ∼= 821 3427 ∼ 839 ∼= 821 3428 ∼ 866 ∼= 866 3429 ∼ 750 ∼= 750 3430 ∼ 1091 ∼= 731 3431 ∼ 1091 ∼= 731 3432 ∼ 957 ∼= 957 3433 ∼ 1091 ∼= 731 3434 ∼ 1091 ∼= 731 3435 ∼ 876 ∼= 876 3436 ∼ 939 ∼= 939 3437 ∼ 858 ∼= 858 3438 ∼ 777 ∼= 777 3439 ∼ 1091 ∼= 731 3440 ∼ 1091 ∼= 731 3441 ∼ 930 ∼= 821 3442 ∼ 1091 ∼= 731 3443 ∼ 1091 ∼= 731 3444 ∼ 849 ∼= 849 3445 ∼ 930 ∼= 821 3446 ∼ 849 ∼= 849 3447 ∼ 768 ∼= 731 3448 ∼ 929 ∼= 929 3449 ∼ 956 ∼= 956 3450 ∼ 821 ∼= 821 3451 ∼ 938 ∼= 938 3452 ∼ 965 ∼= 965 3453 ∼ 821 ∼= 821 3454 ∼ 920 ∼= 920 3455 ∼ 840 ∼= 840 3456 ∼ 741 ∼= 741 3457 ∼ 848 ∼= 750 3458 ∼ 857 ∼= 857 3459 ∼ 839 ∼= 821 3460 ∼ 875 ∼= 875 3461 ∼ 884 ∼= 884 3462 ∼ 866 ∼= 866 3463 ∼ 821 ∼= 821 3464 ∼ 821 ∼= 821 3465 ∼ 750 ∼= 750 3466 ∼ 929 ∼= 929 3467 ∼ 938 ∼= 938 3468 ∼ 920 ∼= 920 3469 ∼ 956 ∼= 956 3470 ∼ 965 ∼= 965 3471 ∼ 840 ∼= 840 3472 ∼ 821 ∼= 821 3473 ∼ 821 ∼= 821 3474 ∼ 741 ∼= 741 3475 ∼ 767 ∼= 731 3476 ∼ 776 ∼= 776 3477 ∼ 740 ∼= 740 3478 ∼ 776 ∼= 776 3479 ∼ 803 ∼= 771 3480 ∼ 749 ∼= 749 3481 ∼ 740 ∼= 740 3482 ∼ 749 ∼= 749 3483 ∼ 731 ∼= 731 3484 ∼ 1091 ∼= 731 3485 ∼ 1091 ∼= 731 3486 ∼ 965 ∼= 965 3487 ∼ 1091 ∼= 731 3488 ∼ 1091 ∼= 731 3489 ∼ 884 ∼= 884 3490 ∼ 965 ∼= 965 3491 ∼ 884 ∼= 884 3492 ∼ 803 ∼= 771 3493 ∼ 1091 ∼= 731 3494 ∼ 1091 ∼= 731 3495 ∼ 938 ∼= 938 3496 ∼ 1091 ∼= 731 3497 ∼ 1091 ∼= 731 3498 ∼ 857 ∼= 857 3499 ∼ 956 ∼= 956 3500 ∼ 875 ∼= 875 3501 ∼ 776 ∼= 776 3502 ∼ 849 ∼= 849 3503 ∼ 876 ∼= 876 3504 ∼ 821 ∼= 821 3505 ∼ 858 ∼= 858 3506 ∼ 885 ∼= 885 3507 ∼ 821 ∼= 821 3508 ∼ 840 ∼= 840 3509 ∼ 866 ∼= 866 3510 ∼ 749 ∼= 749 3511 ∼ 1091 ∼= 731 3512 ∼ 1091 ∼= 731 3513 ∼ 956 ∼= 956 3514 ∼ 1091 ∼= 731 3515 ∼ 1091 ∼= 731 3516 ∼ 875 ∼= 875 3517 ∼ 938 ∼= 938 3518 ∼ 857 ∼= 857 3519 ∼ 776 ∼= 776 3520 ∼ 1091 ∼= 731 3521 ∼ 1091 ∼= 731 3522 ∼ 929 ∼= 929 3523 ∼ 1091 ∼= 731 3524 ∼ 1091 ∼= 731 3525 ∼ 848 ∼= 750 3526 ∼ 929 ∼= 929 3527 ∼ 848 ∼= 750 3528 ∼ 767 ∼= 731 3529 ∼ 930 ∼= 821 3530 ∼ 957 ∼= 957 3531 ∼ 821 ∼= 821 3532 ∼ 939 ∼= 939 3533 ∼ 966 ∼= 966 3534 ∼ 821 ∼= 821 3535 ∼ 920 ∼= 920 3536 ∼ 839 ∼= 821 3537 ∼ 740 ∼= 740 3538 ∼ 849 ∼= 849 3539 ∼ 858 ∼= 858 3540 ∼ 840 ∼= 840 3541 ∼ 876 ∼= 876 3542 ∼ 885 ∼= 885 3543 ∼ 866 ∼= 866 3544 ∼ 821 ∼= 821 3545 ∼ 821 ∼= 821 3546 ∼ 749 ∼= 749 3547 ∼ 930 ∼= 821 3548 ∼ 939 ∼= 939 3549 ∼ 920 ∼= 920 3550 ∼ 957 ∼= 957 3551 ∼ 966 ∼= 966 3552 ∼ 839 ∼= 821 3553 ∼ 821 ∼= 821 3554 ∼ 821 ∼= 821 3555 ∼ 740 ∼= 740 3556 ∼ 768 ∼= 731 3557 ∼ 777 ∼= 777 3558 ∼ 741 ∼= 741 3559 ∼ 777 ∼= 777 3560 ∼ 804 ∼= 731 3561 ∼ 750 ∼= 750 3562 ∼ 741 ∼= 741 3563 ∼ 750 ∼= 750 3564 ∼ 731 ∼= 731 3565 ∼ 1090 ∼= 1090 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 3566 ∼ 1090 ∼= 1090 3567 ∼ 964 ∼= 739 3568 ∼ 1090 ∼= 1090 3569 ∼ 1090 ∼= 1090 3570 ∼ 883 ∼= 883 3571 ∼ 964 ∼= 739 3572 ∼ 883 ∼= 883 3573 ∼ 802 ∼= 802 3574 ∼ 1090 ∼= 1090 3575 ∼ 1090 ∼= 1090 3576 ∼ 937 ∼= 937 3577 ∼ 1090 ∼= 1090 3578 ∼ 1090 ∼= 1090 3579 ∼ 856 ∼= 856 3580 ∼ 955 ∼= 937 3581 ∼ 874 ∼= 874 3582 ∼ 775 ∼= 775 3583 ∼ 847 ∼= 847 3584 ∼ 874 ∼= 874 3585 ∼ 820 ∼= 820 3586 ∼ 856 ∼= 856 3587 ∼ 883 ∼= 883 3588 ∼ 820 ∼= 820 3589 ∼ 838 ∼= 838 3590 ∼ 865 ∼= 820 3591 ∼ 748 ∼= 748 3592 ∼ 1090 ∼= 1090 3593 ∼ 1090 ∼= 1090 3594 ∼ 955 ∼= 937 3595 ∼ 1090 ∼= 1090 3596 ∼ 1090 ∼= 1090 3597 ∼ 874 ∼= 874 3598 ∼ 937 ∼= 937 3599 ∼ 856 ∼= 856 3600 ∼ 775 ∼= 775 3601 ∼ 1090 ∼= 1090 3602 ∼ 1090 ∼= 1090 3603 ∼ 928 ∼= 820 3604 ∼ 1090 ∼= 1090 3605 ∼ 1090 ∼= 1090 3606 ∼ 847 ∼= 847 3607 ∼ 928 ∼= 820 3608 ∼ 847 ∼= 847 3609 ∼ 766 ∼= 730 3610 ∼ 928 ∼= 820 3611 ∼ 955 ∼= 937 3612 ∼ 820 ∼= 820 3613 ∼ 937 ∼= 937 3614 ∼ 964 ∼= 739 3615 ∼ 820 ∼= 820 3616 ∼ 919 ∼= 820 3617 ∼ 838 ∼= 838 3618 ∼ 739 ∼= 739 3619 ∼ 847 ∼= 847 3620 ∼ 856 ∼= 856 3621 ∼ 838 ∼= 838 3622 ∼ 874 ∼= 874 3623 ∼ 883 ∼= 883 3624 ∼ 865 ∼= 820 3625 ∼ 820 ∼= 820 3626 ∼ 820 ∼= 820 3627 ∼ 748 ∼= 748 3628 ∼ 928 ∼= 820 3629 ∼ 937 ∼= 937 3630 ∼ 919 ∼= 820 3631 ∼ 955 ∼= 937 3632 ∼ 964 ∼= 739 3633 ∼ 838 ∼= 838 3634 ∼ 820 ∼= 820 3635 ∼ 820 ∼= 820 3636 ∼ 739 ∼= 739 3637 ∼ 766 ∼= 730 3638 ∼ 775 ∼= 775 3639 ∼ 739 ∼= 739 3640 ∼ 775 ∼= 775 3641 ∼ 802 ∼= 802 3642 ∼ 748 ∼= 748 3643 ∼ 739 ∼= 739 3644 ∼ 748 ∼= 748 3645 ∼ 730 ∼= 730 3646 ∼ 730 ∼= 730 3647 ∼ 2190 ∼= 750 3648 ∼ 730 ∼= 730 3649 ∼ 2190 ∼= 750 3650 ∼ 2196 ∼= 802 3651 ∼ 2193 ∼= 2193 3652 ∼ 730 ∼= 730 3653 ∼ 2193 ∼= 2193 3654 ∼ 730 ∼= 730 3655 ∼ 820 ∼= 820 3656 ∼ 2352 ∼= 740 3657 ∼ 820 ∼= 820 3658 ∼ 2352 ∼= 740 3659 ∼ 2358 ∼= 820 3660 ∼ 2355 ∼= 2355 3661 ∼ 820 ∼= 820 3662 ∼ 2355 ∼= 2355 3663 ∼ 820 ∼= 820 3664 ∼ 730 ∼= 730 3665 ∼ 2271 ∼= 2271 3666 ∼ 730 ∼= 730 3667 ∼ 2271 ∼= 2271 3668 ∼ 2277 ∼= 2277 3669 ∼ 2274 ∼= 2274 3670 ∼ 730 ∼= 730 3671 ∼ 2274 ∼= 2274 3672 ∼ 730 ∼= 730 3673 ∼ 820 ∼= 820 3674 ∼ 2352 ∼= 740 3675 ∼ 820 ∼= 820 3676 ∼ 2352 ∼= 740 3677 ∼ 2358 ∼= 820 3678 ∼ 2355 ∼= 2355 3679 ∼ 820 ∼= 820 3680 ∼ 2355 ∼= 2355 3681 ∼ 820 ∼= 820 3682 ∼ 1090 ∼= 1090 3683 ∼ 2838 ∼= 750 3684 ∼ 1090 ∼= 1090 3685 ∼ 2838 ∼= 750 3686 ∼ 2844 ∼= 730 3687 ∼ 2841 ∼= 2841 3688 ∼ 1090 ∼= 1090 3689 ∼ 2841 ∼= 2841 3690 ∼ 1090 ∼= 1090 3691 ∼ 820 ∼= 820 3692 ∼ 2399 ∼= 2399 3693 ∼ 820 ∼= 820 3694 ∼ 2399 ∼= 2399 3695 ∼ 2426 ∼= 2277 3696 ∼ 2372 ∼= 2372 3697 ∼ 820 ∼= 820 3698 ∼ 2372 ∼= 2372 3699 ∼ 820 ∼= 820 3700 ∼ 730 ∼= 730 3701 ∼ 2271 ∼= 2271 3702 ∼ 730 ∼= 730 3703 ∼ 2271 ∼= 2271 3704 ∼ 2277 ∼= 2277 3705 ∼ 2274 ∼= 2274 3706 ∼ 730 ∼= 730 3707 ∼ 2274 ∼= 2274 3708 ∼ 730 ∼= 730 3709 ∼ 820 ∼= 820 3710 ∼ 2399 ∼= 2399 3711 ∼ 820 ∼= 820 3712 ∼ 2399 ∼= 2399 3713 ∼ 2426 ∼= 2277 3714 ∼ 2372 ∼= 2372 3715 ∼ 820 ∼= 820 3716 ∼ 2372 ∼= 2372 3717 ∼ 820 ∼= 820 3718 ∼ 730 ∼= 730 3719 ∼ 2237 ∼= 2237 3720 ∼ 730 ∼= 730 3721 ∼ 2237 ∼= 2237 3722 ∼ 2264 ∼= 730 3723 ∼ 2210 ∼= 2210 3724 ∼ 730 ∼= 730 3725 ∼ 2210 ∼= 2210 3726 ∼ 730 ∼= 730 3727 ∼ 2206 ∼= 748 3728 ∼ 731 ∼= 731 3729 ∼ 2207 ∼= 2207 3730 ∼ 2212 ∼= 2212 3731 ∼ 2214 ∼= 748 3732 ∼ 2213 ∼= 2213 3733 ∼ 2209 ∼= 2209 38 Classification of groups generated by automata 3734 ∼ 731 ∼= 731 3735 ∼ 2210 ∼= 2210 3736 ∼ 2368 ∼= 739 3737 ∼ 821 ∼= 821 3738 ∼ 2369 ∼= 2369 3739 ∼ 2374 ∼= 821 3740 ∼ 2376 ∼= 739 3741 ∼ 2375 ∼= 2375 3742 ∼ 2371 ∼= 2371 3743 ∼ 821 ∼= 821 3744 ∼ 2372 ∼= 2372 3745 ∼ 2287 ∼= 2287 3746 ∼ 731 ∼= 731 3747 ∼ 2285 ∼= 2285 3748 ∼ 2293 ∼= 2293 3749 ∼ 2295 ∼= 2295 3750 ∼ 2294 ∼= 2294 3751 ∼ 2283 ∼= 2283 3752 ∼ 731 ∼= 731 3753 ∼ 2274 ∼= 2274 3754 ∼ 2368 ∼= 739 3755 ∼ 821 ∼= 821 3756 ∼ 2369 ∼= 2369 3757 ∼ 2374 ∼= 821 3758 ∼ 2376 ∼= 739 3759 ∼ 2375 ∼= 2375 3760 ∼ 2371 ∼= 2371 3761 ∼ 821 ∼= 821 3762 ∼ 2372 ∼= 2372 3763 ∼ 2854 ∼= 847 3764 ∼ 1091 ∼= 731 3765 ∼ 2852 ∼= 849 3766 ∼ 2860 ∼= 2212 3767 ∼ 2862 ∼= 847 3768 ∼ 2861 ∼= 731 3769 ∼ 2850 ∼= 2850 3770 ∼ 1091 ∼= 731 3771 ∼ 2841 ∼= 2841 3772 ∼ 2391 ∼= 2391 3773 ∼ 821 ∼= 821 3774 ∼ 2366 ∼= 2366 3775 ∼ 2402 ∼= 2402 3776 ∼ 2427 ∼= 2427 3777 ∼ 2375 ∼= 2375 3778 ∼ 2364 ∼= 2364 3779 ∼ 821 ∼= 821 3780 ∼ 2355 ∼= 2355 3781 ∼ 2287 ∼= 2287 3782 ∼ 731 ∼= 731 3783 ∼ 2285 ∼= 2285 3784 ∼ 2293 ∼= 2293 3785 ∼ 2295 ∼= 2295 3786 ∼ 2294 ∼= 2294 3787 ∼ 2283 ∼= 2283 3788 ∼ 731 ∼= 731 3789 ∼ 2274 ∼= 2274 3790 ∼ 2391 ∼= 2391 3791 ∼ 821 ∼= 821 3792 ∼ 2366 ∼= 2366 3793 ∼ 2402 ∼= 2402 3794 ∼ 2427 ∼= 2427 3795 ∼ 2375 ∼= 2375 3796 ∼ 2364 ∼= 2364 3797 ∼ 821 ∼= 821 3798 ∼ 2355 ∼= 2355 3799 ∼ 2229 ∼= 2229 3800 ∼ 731 ∼= 731 3801 ∼ 2204 ∼= 2204 3802 ∼ 2240 ∼= 2240 3803 ∼ 2265 ∼= 2265 3804 ∼ 2213 ∼= 2213 3805 ∼ 2202 ∼= 2202 3806 ∼ 731 ∼= 731 3807 ∼ 2193 ∼= 2193 3808 ∼ 730 ∼= 730 3809 ∼ 2199 ∼= 2199 3810 ∼ 730 ∼= 730 3811 ∼ 2203 ∼= 2203 3812 ∼ 2205 ∼= 775 3813 ∼ 2204 ∼= 2204 3814 ∼ 730 ∼= 730 3815 ∼ 2202 ∼= 2202 3816 ∼ 730 ∼= 730 3817 ∼ 820 ∼= 820 3818 ∼ 2361 ∼= 2361 3819 ∼ 820 ∼= 820 3820 ∼ 2365 ∼= 2365 3821 ∼ 2367 ∼= 2367 3822 ∼ 2366 ∼= 2366 3823 ∼ 820 ∼= 820 3824 ∼ 2364 ∼= 2364 3825 ∼ 820 ∼= 820 3826 ∼ 730 ∼= 730 3827 ∼ 2280 ∼= 2280 3828 ∼ 730 ∼= 730 3829 ∼ 2284 ∼= 2284 3830 ∼ 2286 ∼= 2286 3831 ∼ 2285 ∼= 2285 3832 ∼ 730 ∼= 730 3833 ∼ 2283 ∼= 2283 3834 ∼ 730 ∼= 730 3835 ∼ 820 ∼= 820 3836 ∼ 2361 ∼= 2361 3837 ∼ 820 ∼= 820 3838 ∼ 2365 ∼= 2365 3839 ∼ 2367 ∼= 2367 3840 ∼ 2366 ∼= 2366 3841 ∼ 820 ∼= 820 3842 ∼ 2364 ∼= 2364 3843 ∼ 820 ∼= 820 3844 ∼ 1090 ∼= 1090 3845 ∼ 2847 ∼= 929 3846 ∼ 1090 ∼= 1090 3847 ∼ 2851 ∼= 929 3848 ∼ 2853 ∼= 2853 3849 ∼ 2852 ∼= 849 3850 ∼ 1090 ∼= 1090 3851 ∼ 2850 ∼= 2850 3852 ∼ 1090 ∼= 1090 3853 ∼ 820 ∼= 820 3854 ∼ 2398 ∼= 2398 3855 ∼ 820 ∼= 820 3856 ∼ 2396 ∼= 2396 3857 ∼ 2423 ∼= 2423 3858 ∼ 2369 ∼= 2369 3859 ∼ 820 ∼= 820 3860 ∼ 2371 ∼= 2371 3861 ∼ 820 ∼= 820 3862 ∼ 730 ∼= 730 3863 ∼ 2280 ∼= 2280 3864 ∼ 730 ∼= 730 3865 ∼ 2284 ∼= 2284 3866 ∼ 2286 ∼= 2286 3867 ∼ 2285 ∼= 2285 3868 ∼ 730 ∼= 730 3869 ∼ 2283 ∼= 2283 3870 ∼ 730 ∼= 730 3871 ∼ 820 ∼= 820 3872 ∼ 2398 ∼= 2398 3873 ∼ 820 ∼= 820 3874 ∼ 2396 ∼= 2396 3875 ∼ 2423 ∼= 2423 3876 ∼ 2369 ∼= 2369 3877 ∼ 820 ∼= 820 3878 ∼ 2371 ∼= 2371 3879 ∼ 820 ∼= 820 3880 ∼ 730 ∼= 730 3881 ∼ 2236 ∼= 2236 3882 ∼ 730 ∼= 730 3883 ∼ 2234 ∼= 2234 3884 ∼ 2261 ∼= 2261 3885 ∼ 2207 ∼= 2207 3886 ∼ 730 ∼= 730 3887 ∼ 2209 ∼= 2209 3888 ∼ 730 ∼= 730 3889 ∼ 2206 ∼= 748 3890 ∼ 2212 ∼= 2212 3891 ∼ 2209 ∼= 2209 3892 ∼ 731 ∼= 731 3893 ∼ 2214 ∼= 748 3894 ∼ 731 ∼= 731 3895 ∼ 2207 ∼= 2207 3896 ∼ 2213 ∼= 2213 3897 ∼ 2210 ∼= 2210 3898 ∼ 2368 ∼= 739 3899 ∼ 2374 ∼= 821 3900 ∼ 2371 ∼= 2371 3901 ∼ 821 ∼= 821 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 3902 ∼ 2376 ∼= 739 3903 ∼ 821 ∼= 821 3904 ∼ 2369 ∼= 2369 3905 ∼ 2375 ∼= 2375 3906 ∼ 2372 ∼= 2372 3907 ∼ 2287 ∼= 2287 3908 ∼ 2293 ∼= 2293 3909 ∼ 2283 ∼= 2283 3910 ∼ 731 ∼= 731 3911 ∼ 2295 ∼= 2295 3912 ∼ 731 ∼= 731 3913 ∼ 2285 ∼= 2285 3914 ∼ 2294 ∼= 2294 3915 ∼ 2274 ∼= 2274 3916 ∼ 2368 ∼= 739 3917 ∼ 2374 ∼= 821 3918 ∼ 2371 ∼= 2371 3919 ∼ 821 ∼= 821 3920 ∼ 2376 ∼= 739 3921 ∼ 821 ∼= 821 3922 ∼ 2369 ∼= 2369 3923 ∼ 2375 ∼= 2375 3924 ∼ 2372 ∼= 2372 3925 ∼ 2854 ∼= 847 3926 ∼ 2860 ∼= 2212 3927 ∼ 2850 ∼= 2850 3928 ∼ 1091 ∼= 731 3929 ∼ 2862 ∼= 847 3930 ∼ 1091 ∼= 731 3931 ∼ 2852 ∼= 849 3932 ∼ 2861 ∼= 731 3933 ∼ 2841 ∼= 2841 3934 ∼ 2391 ∼= 2391 3935 ∼ 2402 ∼= 2402 3936 ∼ 2364 ∼= 2364 3937 ∼ 821 ∼= 821 3938 ∼ 2427 ∼= 2427 3939 ∼ 821 ∼= 821 3940 ∼ 2366 ∼= 2366 3941 ∼ 2375 ∼= 2375 3942 ∼ 2355 ∼= 2355 3943 ∼ 2287 ∼= 2287 3944 ∼ 2293 ∼= 2293 3945 ∼ 2283 ∼= 2283 3946 ∼ 731 ∼= 731 3947 ∼ 2295 ∼= 2295 3948 ∼ 731 ∼= 731 3949 ∼ 2285 ∼= 2285 3950 ∼ 2294 ∼= 2294 3951 ∼ 2274 ∼= 2274 3952 ∼ 2391 ∼= 2391 3953 ∼ 2402 ∼= 2402 3954 ∼ 2364 ∼= 2364 3955 ∼ 821 ∼= 821 3956 ∼ 2427 ∼= 2427 3957 ∼ 821 ∼= 821 3958 ∼ 2366 ∼= 2366 3959 ∼ 2375 ∼= 2375 3960 ∼ 2355 ∼= 2355 3961 ∼ 2229 ∼= 2229 3962 ∼ 2240 ∼= 2240 3963 ∼ 2202 ∼= 2202 3964 ∼ 731 ∼= 731 3965 ∼ 2265 ∼= 2265 3966 ∼ 731 ∼= 731 3967 ∼ 2204 ∼= 2204 3968 ∼ 2213 ∼= 2213 3969 ∼ 2193 ∼= 2193 3970 ∼ 2260 ∼= 802 3971 ∼ 2262 ∼= 750 3972 ∼ 2261 ∼= 2261 3973 ∼ 2262 ∼= 750 3974 ∼ 734 ∼= 730 3975 ∼ 2265 ∼= 2265 3976 ∼ 2261 ∼= 2261 3977 ∼ 2265 ∼= 2265 3978 ∼ 2264 ∼= 730 3979 ∼ 2422 ∼= 820 3980 ∼ 2424 ∼= 966 3981 ∼ 2423 ∼= 2423 3982 ∼ 2424 ∼= 966 3983 ∼ 824 ∼= 820 3984 ∼ 2427 ∼= 2427 3985 ∼ 2423 ∼= 2423 3986 ∼ 2427 ∼= 2427 3987 ∼ 2426 ∼= 2277 3988 ∼ 2313 ∼= 2277 3989 ∼ 2322 ∼= 2322 3990 ∼ 2286 ∼= 2286 3991 ∼ 2322 ∼= 2322 3992 ∼ 734 ∼= 730 3993 ∼ 2295 ∼= 2295 3994 ∼ 2286 ∼= 2286 3995 ∼ 2295 ∼= 2295 3996 ∼ 2277 ∼= 2277 3997 ∼ 2422 ∼= 820 3998 ∼ 2424 ∼= 966 3999 ∼ 2423 ∼= 2423 4000 ∼ 2424 ∼= 966 4001 ∼ 824 ∼= 820 4002 ∼ 2427 ∼= 2427 4003 ∼ 2423 ∼= 2423 4004 ∼ 2427 ∼= 2427 4005 ∼ 2426 ∼= 2277 4006 ∼ 2880 ∼= 730 4007 ∼ 2889 ∼= 750 4008 ∼ 2853 ∼= 2853 4009 ∼ 2889 ∼= 750 4010 ∼ 1094 ∼= 1090 4011 ∼ 2862 ∼= 847 4012 ∼ 2853 ∼= 2853 4013 ∼ 2862 ∼= 847 4014 ∼ 2844 ∼= 730 4015 ∼ 2394 ∼= 820 4016 ∼ 2403 ∼= 2287 4017 ∼ 2367 ∼= 2367 4018 ∼ 2403 ∼= 2287 4019 ∼ 824 ∼= 820 4020 ∼ 2376 ∼= 739 4021 ∼ 2367 ∼= 2367 4022 ∼ 2376 ∼= 739 4023 ∼ 2358 ∼= 820 4024 ∼ 2313 ∼= 2277 4025 ∼ 2322 ∼= 2322 4026 ∼ 2286 ∼= 2286 4027 ∼ 2322 ∼= 2322 4028 ∼ 734 ∼= 730 4029 ∼ 2295 ∼= 2295 4030 ∼ 2286 ∼= 2286 4031 ∼ 2295 ∼= 2295 4032 ∼ 2277 ∼= 2277 4033 ∼ 2394 ∼= 820 4034 ∼ 2403 ∼= 2287 4035 ∼ 2367 ∼= 2367 4036 ∼ 2403 ∼= 2287 4037 ∼ 824 ∼= 820 4038 ∼ 2376 ∼= 739 4039 ∼ 2367 ∼= 2367 4040 ∼ 2376 ∼= 739 4041 ∼ 2358 ∼= 820 4042 ∼ 2232 ∼= 730 4043 ∼ 2241 ∼= 739 4044 ∼ 2205 ∼= 775 4045 ∼ 2241 ∼= 739 4046 ∼ 734 ∼= 730 4047 ∼ 2214 ∼= 748 4048 ∼ 2205 ∼= 775 4049 ∼ 2214 ∼= 748 4050 ∼ 2196 ∼= 802 4051 ∼ 2233 ∼= 2233 4052 ∼ 2239 ∼= 2239 4053 ∼ 2236 ∼= 2236 4054 ∼ 731 ∼= 731 4055 ∼ 2241 ∼= 739 4056 ∼ 731 ∼= 731 4057 ∼ 2234 ∼= 2234 4058 ∼ 2240 ∼= 2240 4059 ∼ 2237 ∼= 2237 4060 ∼ 2395 ∼= 2395 4061 ∼ 2401 ∼= 2401 4062 ∼ 2398 ∼= 2398 4063 ∼ 821 ∼= 821 4064 ∼ 2403 ∼= 2287 4065 ∼ 821 ∼= 821 4066 ∼ 2396 ∼= 2396 4067 ∼ 2402 ∼= 2402 4068 ∼ 2399 ∼= 2399 4069 ∼ 2307 ∼= 2307 40 Classification of groups generated by automata 4070 ∼ 2320 ∼= 2294 4071 ∼ 2280 ∼= 2280 4072 ∼ 731 ∼= 731 4073 ∼ 2322 ∼= 2322 4074 ∼ 731 ∼= 731 4075 ∼ 2284 ∼= 2284 4076 ∼ 2293 ∼= 2293 4077 ∼ 2271 ∼= 2271 4078 ∼ 2395 ∼= 2395 4079 ∼ 2401 ∼= 2401 4080 ∼ 2398 ∼= 2398 4081 ∼ 821 ∼= 821 4082 ∼ 2403 ∼= 2287 4083 ∼ 821 ∼= 821 4084 ∼ 2396 ∼= 2396 4085 ∼ 2402 ∼= 2402 4086 ∼ 2399 ∼= 2399 4087 ∼ 2874 ∼= 820 4088 ∼ 2887 ∼= 731 4089 ∼ 2847 ∼= 929 4090 ∼ 1091 ∼= 731 4091 ∼ 2889 ∼= 750 4092 ∼ 1091 ∼= 731 4093 ∼ 2851 ∼= 929 4094 ∼ 2860 ∼= 2212 4095 ∼ 2838 ∼= 750 4096 ∼ 2388 ∼= 821 4097 ∼ 2401 ∼= 2401 4098 ∼ 2361 ∼= 2361 4099 ∼ 821 ∼= 821 4100 ∼ 2424 ∼= 966 4101 ∼ 821 ∼= 821 4102 ∼ 2365 ∼= 2365 4103 ∼ 2374 ∼= 821 4104 ∼ 2352 ∼= 740 4105 ∼ 2307 ∼= 2307 4106 ∼ 2320 ∼= 2294 4107 ∼ 2280 ∼= 2280 4108 ∼ 731 ∼= 731 4109 ∼ 2322 ∼= 2322 4110 ∼ 731 ∼= 731 4111 ∼ 2284 ∼= 2284 4112 ∼ 2293 ∼= 2293 4113 ∼ 2271 ∼= 2271 4114 ∼ 2388 ∼= 821 4115 ∼ 2401 ∼= 2401 4116 ∼ 2361 ∼= 2361 4117 ∼ 821 ∼= 821 4118 ∼ 2424 ∼= 966 4119 ∼ 821 ∼= 821 4120 ∼ 2365 ∼= 2365 4121 ∼ 2374 ∼= 821 4122 ∼ 2352 ∼= 740 4123 ∼ 2226 ∼= 820 4124 ∼ 2239 ∼= 2239 4125 ∼ 2199 ∼= 2199 4126 ∼ 731 ∼= 731 4127 ∼ 2262 ∼= 750 4128 ∼ 731 ∼= 731 4129 ∼ 2203 ∼= 2203 4130 ∼ 2212 ∼= 2212 4131 ∼ 2190 ∼= 750 4132 ∼ 730 ∼= 730 4133 ∼ 2203 ∼= 2203 4134 ∼ 730 ∼= 730 4135 ∼ 2199 ∼= 2199 4136 ∼ 2205 ∼= 775 4137 ∼ 2202 ∼= 2202 4138 ∼ 730 ∼= 730 4139 ∼ 2204 ∼= 2204 4140 ∼ 730 ∼= 730 4141 ∼ 820 ∼= 820 4142 ∼ 2365 ∼= 2365 4143 ∼ 820 ∼= 820 4144 ∼ 2361 ∼= 2361 4145 ∼ 2367 ∼= 2367 4146 ∼ 2364 ∼= 2364 4147 ∼ 820 ∼= 820 4148 ∼ 2366 ∼= 2366 4149 ∼ 820 ∼= 820 4150 ∼ 730 ∼= 730 4151 ∼ 2284 ∼= 2284 4152 ∼ 730 ∼= 730 4153 ∼ 2280 ∼= 2280 4154 ∼ 2286 ∼= 2286 4155 ∼ 2283 ∼= 2283 4156 ∼ 730 ∼= 730 4157 ∼ 2285 ∼= 2285 4158 ∼ 730 ∼= 730 4159 ∼ 820 ∼= 820 4160 ∼ 2365 ∼= 2365 4161 ∼ 820 ∼= 820 4162 ∼ 2361 ∼= 2361 4163 ∼ 2367 ∼= 2367 4164 ∼ 2364 ∼= 2364 4165 ∼ 820 ∼= 820 4166 ∼ 2366 ∼= 2366 4167 ∼ 820 ∼= 820 4168 ∼ 1090 ∼= 1090 4169 ∼ 2851 ∼= 929 4170 ∼ 1090 ∼= 1090 4171 ∼ 2847 ∼= 929 4172 ∼ 2853 ∼= 2853 4173 ∼ 2850 ∼= 2850 4174 ∼ 1090 ∼= 1090 4175 ∼ 2852 ∼= 849 4176 ∼ 1090 ∼= 1090 4177 ∼ 820 ∼= 820 4178 ∼ 2396 ∼= 2396 4179 ∼ 820 ∼= 820 4180 ∼ 2398 ∼= 2398 4181 ∼ 2423 ∼= 2423 4182 ∼ 2371 ∼= 2371 4183 ∼ 820 ∼= 820 4184 ∼ 2369 ∼= 2369 4185 ∼ 820 ∼= 820 4186 ∼ 730 ∼= 730 4187 ∼ 2284 ∼= 2284 4188 ∼ 730 ∼= 730 4189 ∼ 2280 ∼= 2280 4190 ∼ 2286 ∼= 2286 4191 ∼ 2283 ∼= 2283 4192 ∼ 730 ∼= 730 4193 ∼ 2285 ∼= 2285 4194 ∼ 730 ∼= 730 4195 ∼ 820 ∼= 820 4196 ∼ 2396 ∼= 2396 4197 ∼ 820 ∼= 820 4198 ∼ 2398 ∼= 2398 4199 ∼ 2423 ∼= 2423 4200 ∼ 2371 ∼= 2371 4201 ∼ 820 ∼= 820 4202 ∼ 2369 ∼= 2369 4203 ∼ 820 ∼= 820 4204 ∼ 730 ∼= 730 4205 ∼ 2234 ∼= 2234 4206 ∼ 730 ∼= 730 4207 ∼ 2236 ∼= 2236 4208 ∼ 2261 ∼= 2261 4209 ∼ 2209 ∼= 2209 4210 ∼ 730 ∼= 730 4211 ∼ 2207 ∼= 2207 4212 ∼ 730 ∼= 730 4213 ∼ 2233 ∼= 2233 4214 ∼ 731 ∼= 731 4215 ∼ 2234 ∼= 2234 4216 ∼ 2239 ∼= 2239 4217 ∼ 2241 ∼= 739 4218 ∼ 2240 ∼= 2240 4219 ∼ 2236 ∼= 2236 4220 ∼ 731 ∼= 731 4221 ∼ 2237 ∼= 2237 4222 ∼ 2395 ∼= 2395 4223 ∼ 821 ∼= 821 4224 ∼ 2396 ∼= 2396 4225 ∼ 2401 ∼= 2401 4226 ∼ 2403 ∼= 2287 4227 ∼ 2402 ∼= 2402 4228 ∼ 2398 ∼= 2398 4229 ∼ 821 ∼= 821 4230 ∼ 2399 ∼= 2399 4231 ∼ 2307 ∼= 2307 4232 ∼ 731 ∼= 731 4233 ∼ 2284 ∼= 2284 4234 ∼ 2320 ∼= 2294 4235 ∼ 2322 ∼= 2322 4236 ∼ 2293 ∼= 2293 4237 ∼ 2280 ∼= 2280 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 4238 ∼ 731 ∼= 731 4239 ∼ 2271 ∼= 2271 4240 ∼ 2395 ∼= 2395 4241 ∼ 821 ∼= 821 4242 ∼ 2396 ∼= 2396 4243 ∼ 2401 ∼= 2401 4244 ∼ 2403 ∼= 2287 4245 ∼ 2402 ∼= 2402 4246 ∼ 2398 ∼= 2398 4247 ∼ 821 ∼= 821 4248 ∼ 2399 ∼= 2399 4249 ∼ 2874 ∼= 820 4250 ∼ 1091 ∼= 731 4251 ∼ 2851 ∼= 929 4252 ∼ 2887 ∼= 731 4253 ∼ 2889 ∼= 750 4254 ∼ 2860 ∼= 2212 4255 ∼ 2847 ∼= 929 4256 ∼ 1091 ∼= 731 4257 ∼ 2838 ∼= 750 4258 ∼ 2388 ∼= 821 4259 ∼ 821 ∼= 821 4260 ∼ 2365 ∼= 2365 4261 ∼ 2401 ∼= 2401 4262 ∼ 2424 ∼= 966 4263 ∼ 2374 ∼= 821 4264 ∼ 2361 ∼= 2361 4265 ∼ 821 ∼= 821 4266 ∼ 2352 ∼= 740 4267 ∼ 2307 ∼= 2307 4268 ∼ 731 ∼= 731 4269 ∼ 2284 ∼= 2284 4270 ∼ 2320 ∼= 2294 4271 ∼ 2322 ∼= 2322 4272 ∼ 2293 ∼= 2293 4273 ∼ 2280 ∼= 2280 4274 ∼ 731 ∼= 731 4275 ∼ 2271 ∼= 2271 4276 ∼ 2388 ∼= 821 4277 ∼ 821 ∼= 821 4278 ∼ 2365 ∼= 2365 4279 ∼ 2401 ∼= 2401 4280 ∼ 2424 ∼= 966 4281 ∼ 2374 ∼= 821 4282 ∼ 2361 ∼= 2361 4283 ∼ 821 ∼= 821 4284 ∼ 2352 ∼= 740 4285 ∼ 2226 ∼= 820 4286 ∼ 731 ∼= 731 4287 ∼ 2203 ∼= 2203 4288 ∼ 2239 ∼= 2239 4289 ∼ 2262 ∼= 750 4290 ∼ 2212 ∼= 2212 4291 ∼ 2199 ∼= 2199 4292 ∼ 731 ∼= 731 4293 ∼ 2190 ∼= 750 4294 ∼ 730 ∼= 730 4295 ∼ 2226 ∼= 820 4296 ∼ 730 ∼= 730 4297 ∼ 2226 ∼= 820 4298 ∼ 2232 ∼= 730 4299 ∼ 2229 ∼= 2229 4300 ∼ 730 ∼= 730 4301 ∼ 2229 ∼= 2229 4302 ∼ 730 ∼= 730 4303 ∼ 820 ∼= 820 4304 ∼ 2388 ∼= 821 4305 ∼ 820 ∼= 820 4306 ∼ 2388 ∼= 821 4307 ∼ 2394 ∼= 820 4308 ∼ 2391 ∼= 2391 4309 ∼ 820 ∼= 820 4310 ∼ 2391 ∼= 2391 4311 ∼ 820 ∼= 820 4312 ∼ 730 ∼= 730 4313 ∼ 2307 ∼= 2307 4314 ∼ 730 ∼= 730 4315 ∼ 2307 ∼= 2307 4316 ∼ 2313 ∼= 2277 4317 ∼ 2287 ∼= 2287 4318 ∼ 730 ∼= 730 4319 ∼ 2287 ∼= 2287 4320 ∼ 730 ∼= 730 4321 ∼ 820 ∼= 820 4322 ∼ 2388 ∼= 821 4323 ∼ 820 ∼= 820 4324 ∼ 2388 ∼= 821 4325 ∼ 2394 ∼= 820 4326 ∼ 2391 ∼= 2391 4327 ∼ 820 ∼= 820 4328 ∼ 2391 ∼= 2391 4329 ∼ 820 ∼= 820 4330 ∼ 1090 ∼= 1090 4331 ∼ 2874 ∼= 820 4332 ∼ 1090 ∼= 1090 4333 ∼ 2874 ∼= 820 4334 ∼ 2880 ∼= 730 4335 ∼ 2854 ∼= 847 4336 ∼ 1090 ∼= 1090 4337 ∼ 2854 ∼= 847 4338 ∼ 1090 ∼= 1090 4339 ∼ 820 ∼= 820 4340 ∼ 2395 ∼= 2395 4341 ∼ 820 ∼= 820 4342 ∼ 2395 ∼= 2395 4343 ∼ 2422 ∼= 820 4344 ∼ 2368 ∼= 739 4345 ∼ 820 ∼= 820 4346 ∼ 2368 ∼= 739 4347 ∼ 820 ∼= 820 4348 ∼ 730 ∼= 730 4349 ∼ 2307 ∼= 2307 4350 ∼ 730 ∼= 730 4351 ∼ 2307 ∼= 2307 4352 ∼ 2313 ∼= 2277 4353 ∼ 2287 ∼= 2287 4354 ∼ 730 ∼= 730 4355 ∼ 2287 ∼= 2287 4356 ∼ 730 ∼= 730 4357 ∼ 820 ∼= 820 4358 ∼ 2395 ∼= 2395 4359 ∼ 820 ∼= 820 4360 ∼ 2395 ∼= 2395 4361 ∼ 2422 ∼= 820 4362 ∼ 2368 ∼= 739 4363 ∼ 820 ∼= 820 4364 ∼ 2368 ∼= 739 4365 ∼ 820 ∼= 820 4366 ∼ 730 ∼= 730 4367 ∼ 2233 ∼= 2233 4368 ∼ 730 ∼= 730 4369 ∼ 2233 ∼= 2233 4370 ∼ 2260 ∼= 802 4371 ∼ 2206 ∼= 748 4372 ∼ 730 ∼= 730 4373 ∼ 2206 ∼= 748 4374 ∼ 730 ∼= 730 4375 ∼ 1094 ∼= 1090 4376 ∼ 824 ∼= 820 4377 ∼ 824 ∼= 820 4378 ∼ 824 ∼= 820 4379 ∼ 734 ∼= 730 4380 ∼ 734 ∼= 730 4381 ∼ 824 ∼= 820 4382 ∼ 734 ∼= 730 4383 ∼ 734 ∼= 730 4384 ∼ 2889 ∼= 750 4385 ∼ 2424 ∼= 966 4386 ∼ 2403 ∼= 2287 4387 ∼ 2424 ∼= 966 4388 ∼ 2262 ∼= 750 4389 ∼ 2322 ∼= 2322 4390 ∼ 2403 ∼= 2287 4391 ∼ 2322 ∼= 2322 4392 ∼ 2241 ∼= 739 4393 ∼ 2862 ∼= 847 4394 ∼ 2427 ∼= 2427 4395 ∼ 2376 ∼= 739 4396 ∼ 2427 ∼= 2427 4397 ∼ 2265 ∼= 2265 4398 ∼ 2295 ∼= 2295 4399 ∼ 2376 ∼= 739 4400 ∼ 2295 ∼= 2295 4401 ∼ 2214 ∼= 748 4402 ∼ 2889 ∼= 750 4403 ∼ 2424 ∼= 966 4404 ∼ 2403 ∼= 2287 4405 ∼ 2424 ∼= 966 42 Classification of groups generated by automata 4406 ∼ 2262 ∼= 750 4407 ∼ 2322 ∼= 2322 4408 ∼ 2403 ∼= 2287 4409 ∼ 2322 ∼= 2322 4410 ∼ 2241 ∼= 739 4411 ∼ 2880 ∼= 730 4412 ∼ 2422 ∼= 820 4413 ∼ 2394 ∼= 820 4414 ∼ 2422 ∼= 820 4415 ∼ 2260 ∼= 802 4416 ∼ 2313 ∼= 2277 4417 ∼ 2394 ∼= 820 4418 ∼ 2313 ∼= 2277 4419 ∼ 2232 ∼= 730 4420 ∼ 2853 ∼= 2853 4421 ∼ 2423 ∼= 2423 4422 ∼ 2367 ∼= 2367 4423 ∼ 2423 ∼= 2423 4424 ∼ 2261 ∼= 2261 4425 ∼ 2286 ∼= 2286 4426 ∼ 2367 ∼= 2367 4427 ∼ 2286 ∼= 2286 4428 ∼ 2205 ∼= 775 4429 ∼ 2862 ∼= 847 4430 ∼ 2427 ∼= 2427 4431 ∼ 2376 ∼= 739 4432 ∼ 2427 ∼= 2427 4433 ∼ 2265 ∼= 2265 4434 ∼ 2295 ∼= 2295 4435 ∼ 2376 ∼= 739 4436 ∼ 2295 ∼= 2295 4437 ∼ 2214 ∼= 748 4438 ∼ 2853 ∼= 2853 4439 ∼ 2423 ∼= 2423 4440 ∼ 2367 ∼= 2367 4441 ∼ 2423 ∼= 2423 4442 ∼ 2261 ∼= 2261 4443 ∼ 2286 ∼= 2286 4444 ∼ 2367 ∼= 2367 4445 ∼ 2286 ∼= 2286 4446 ∼ 2205 ∼= 775 4447 ∼ 2844 ∼= 730 4448 ∼ 2426 ∼= 2277 4449 ∼ 2358 ∼= 820 4450 ∼ 2426 ∼= 2277 4451 ∼ 2264 ∼= 730 4452 ∼ 2277 ∼= 2277 4453 ∼ 2358 ∼= 820 4454 ∼ 2277 ∼= 2277 4455 ∼ 2196 ∼= 802 4456 ∼ 2862 ∼= 847 4457 ∼ 2376 ∼= 739 4458 ∼ 2427 ∼= 2427 4459 ∼ 2376 ∼= 739 4460 ∼ 2214 ∼= 748 4461 ∼ 2295 ∼= 2295 4462 ∼ 2427 ∼= 2427 4463 ∼ 2295 ∼= 2295 4464 ∼ 2265 ∼= 2265 4465 ∼ 1091 ∼= 731 4466 ∼ 821 ∼= 821 4467 ∼ 821 ∼= 821 4468 ∼ 821 ∼= 821 4469 ∼ 731 ∼= 731 4470 ∼ 731 ∼= 731 4471 ∼ 821 ∼= 821 4472 ∼ 731 ∼= 731 4473 ∼ 731 ∼= 731 4474 ∼ 1091 ∼= 731 4475 ∼ 821 ∼= 821 4476 ∼ 821 ∼= 821 4477 ∼ 821 ∼= 821 4478 ∼ 731 ∼= 731 4479 ∼ 731 ∼= 731 4480 ∼ 821 ∼= 821 4481 ∼ 731 ∼= 731 4482 ∼ 731 ∼= 731 4483 ∼ 2860 ∼= 2212 4484 ∼ 2374 ∼= 821 4485 ∼ 2402 ∼= 2402 4486 ∼ 2374 ∼= 821 4487 ∼ 2212 ∼= 2212 4488 ∼ 2293 ∼= 2293 4489 ∼ 2402 ∼= 2402 4490 ∼ 2293 ∼= 2293 4491 ∼ 2240 ∼= 2240 4492 ∼ 2854 ∼= 847 4493 ∼ 2368 ∼= 739 4494 ∼ 2391 ∼= 2391 4495 ∼ 2368 ∼= 739 4496 ∼ 2206 ∼= 748 4497 ∼ 2287 ∼= 2287 4498 ∼ 2391 ∼= 2391 4499 ∼ 2287 ∼= 2287 4500 ∼ 2229 ∼= 2229 4501 ∼ 2850 ∼= 2850 4502 ∼ 2371 ∼= 2371 4503 ∼ 2364 ∼= 2364 4504 ∼ 2371 ∼= 2371 4505 ∼ 2209 ∼= 2209 4506 ∼ 2283 ∼= 2283 4507 ∼ 2364 ∼= 2364 4508 ∼ 2283 ∼= 2283 4509 ∼ 2202 ∼= 2202 4510 ∼ 2861 ∼= 731 4511 ∼ 2375 ∼= 2375 4512 ∼ 2375 ∼= 2375 4513 ∼ 2375 ∼= 2375 4514 ∼ 2213 ∼= 2213 4515 ∼ 2294 ∼= 2294 4516 ∼ 2375 ∼= 2375 4517 ∼ 2294 ∼= 2294 4518 ∼ 2213 ∼= 2213 4519 ∼ 2852 ∼= 849 4520 ∼ 2369 ∼= 2369 4521 ∼ 2366 ∼= 2366 4522 ∼ 2369 ∼= 2369 4523 ∼ 2207 ∼= 2207 4524 ∼ 2285 ∼= 2285 4525 ∼ 2366 ∼= 2366 4526 ∼ 2285 ∼= 2285 4527 ∼ 2204 ∼= 2204 4528 ∼ 2841 ∼= 2841 4529 ∼ 2372 ∼= 2372 4530 ∼ 2355 ∼= 2355 4531 ∼ 2372 ∼= 2372 4532 ∼ 2210 ∼= 2210 4533 ∼ 2274 ∼= 2274 4534 ∼ 2355 ∼= 2355 4535 ∼ 2274 ∼= 2274 4536 ∼ 2193 ∼= 2193 4537 ∼ 2889 ∼= 750 4538 ∼ 2403 ∼= 2287 4539 ∼ 2424 ∼= 966 4540 ∼ 2403 ∼= 2287 4541 ∼ 2241 ∼= 739 4542 ∼ 2322 ∼= 2322 4543 ∼ 2424 ∼= 966 4544 ∼ 2322 ∼= 2322 4545 ∼ 2262 ∼= 750 4546 ∼ 1091 ∼= 731 4547 ∼ 821 ∼= 821 4548 ∼ 821 ∼= 821 4549 ∼ 821 ∼= 821 4550 ∼ 731 ∼= 731 4551 ∼ 731 ∼= 731 4552 ∼ 821 ∼= 821 4553 ∼ 731 ∼= 731 4554 ∼ 731 ∼= 731 4555 ∼ 1091 ∼= 731 4556 ∼ 821 ∼= 821 4557 ∼ 821 ∼= 821 4558 ∼ 821 ∼= 821 4559 ∼ 731 ∼= 731 4560 ∼ 731 ∼= 731 4561 ∼ 821 ∼= 821 4562 ∼ 731 ∼= 731 4563 ∼ 731 ∼= 731 4564 ∼ 2887 ∼= 731 4565 ∼ 2401 ∼= 2401 4566 ∼ 2401 ∼= 2401 4567 ∼ 2401 ∼= 2401 4568 ∼ 2239 ∼= 2239 4569 ∼ 2320 ∼= 2294 4570 ∼ 2401 ∼= 2401 4571 ∼ 2320 ∼= 2294 4572 ∼ 2239 ∼= 2239 4573 ∼ 2874 ∼= 820 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 4574 ∼ 2395 ∼= 2395 4575 ∼ 2388 ∼= 821 4576 ∼ 2395 ∼= 2395 4577 ∼ 2233 ∼= 2233 4578 ∼ 2307 ∼= 2307 4579 ∼ 2388 ∼= 821 4580 ∼ 2307 ∼= 2307 4581 ∼ 2226 ∼= 820 4582 ∼ 2847 ∼= 929 4583 ∼ 2398 ∼= 2398 4584 ∼ 2361 ∼= 2361 4585 ∼ 2398 ∼= 2398 4586 ∼ 2236 ∼= 2236 4587 ∼ 2280 ∼= 2280 4588 ∼ 2361 ∼= 2361 4589 ∼ 2280 ∼= 2280 4590 ∼ 2199 ∼= 2199 4591 ∼ 2860 ∼= 2212 4592 ∼ 2402 ∼= 2402 4593 ∼ 2374 ∼= 821 4594 ∼ 2402 ∼= 2402 4595 ∼ 2240 ∼= 2240 4596 ∼ 2293 ∼= 2293 4597 ∼ 2374 ∼= 821 4598 ∼ 2293 ∼= 2293 4599 ∼ 2212 ∼= 2212 4600 ∼ 2851 ∼= 929 4601 ∼ 2396 ∼= 2396 4602 ∼ 2365 ∼= 2365 4603 ∼ 2396 ∼= 2396 4604 ∼ 2234 ∼= 2234 4605 ∼ 2284 ∼= 2284 4606 ∼ 2365 ∼= 2365 4607 ∼ 2284 ∼= 2284 4608 ∼ 2203 ∼= 2203 4609 ∼ 2838 ∼= 750 4610 ∼ 2399 ∼= 2399 4611 ∼ 2352 ∼= 740 4612 ∼ 2399 ∼= 2399 4613 ∼ 2237 ∼= 2237 4614 ∼ 2271 ∼= 2271 4615 ∼ 2352 ∼= 740 4616 ∼ 2271 ∼= 2271 4617 ∼ 2190 ∼= 750 4618 ∼ 2862 ∼= 847 4619 ∼ 2376 ∼= 739 4620 ∼ 2427 ∼= 2427 4621 ∼ 2376 ∼= 739 4622 ∼ 2214 ∼= 748 4623 ∼ 2295 ∼= 2295 4624 ∼ 2427 ∼= 2427 4625 ∼ 2295 ∼= 2295 4626 ∼ 2265 ∼= 2265 4627 ∼ 2860 ∼= 2212 4628 ∼ 2374 ∼= 821 4629 ∼ 2402 ∼= 2402 4630 ∼ 2374 ∼= 821 4631 ∼ 2212 ∼= 2212 4632 ∼ 2293 ∼= 2293 4633 ∼ 2402 ∼= 2402 4634 ∼ 2293 ∼= 2293 4635 ∼ 2240 ∼= 2240 4636 ∼ 2861 ∼= 731 4637 ∼ 2375 ∼= 2375 4638 ∼ 2375 ∼= 2375 4639 ∼ 2375 ∼= 2375 4640 ∼ 2213 ∼= 2213 4641 ∼ 2294 ∼= 2294 4642 ∼ 2375 ∼= 2375 4643 ∼ 2294 ∼= 2294 4644 ∼ 2213 ∼= 2213 4645 ∼ 1091 ∼= 731 4646 ∼ 821 ∼= 821 4647 ∼ 821 ∼= 821 4648 ∼ 821 ∼= 821 4649 ∼ 731 ∼= 731 4650 ∼ 731 ∼= 731 4651 ∼ 821 ∼= 821 4652 ∼ 731 ∼= 731 4653 ∼ 731 ∼= 731 4654 ∼ 2854 ∼= 847 4655 ∼ 2368 ∼= 739 4656 ∼ 2391 ∼= 2391 4657 ∼ 2368 ∼= 739 4658 ∼ 2206 ∼= 748 4659 ∼ 2287 ∼= 2287 4660 ∼ 2391 ∼= 2391 4661 ∼ 2287 ∼= 2287 4662 ∼ 2229 ∼= 2229 4663 ∼ 2852 ∼= 849 4664 ∼ 2369 ∼= 2369 4665 ∼ 2366 ∼= 2366 4666 ∼ 2369 ∼= 2369 4667 ∼ 2207 ∼= 2207 4668 ∼ 2285 ∼= 2285 4669 ∼ 2366 ∼= 2366 4670 ∼ 2285 ∼= 2285 4671 ∼ 2204 ∼= 2204 4672 ∼ 1091 ∼= 731 4673 ∼ 821 ∼= 821 4674 ∼ 821 ∼= 821 4675 ∼ 821 ∼= 821 4676 ∼ 731 ∼= 731 4677 ∼ 731 ∼= 731 4678 ∼ 821 ∼= 821 4679 ∼ 731 ∼= 731 4680 ∼ 731 ∼= 731 4681 ∼ 2850 ∼= 2850 4682 ∼ 2371 ∼= 2371 4683 ∼ 2364 ∼= 2364 4684 ∼ 2371 ∼= 2371 4685 ∼ 2209 ∼= 2209 4686 ∼ 2283 ∼= 2283 4687 ∼ 2364 ∼= 2364 4688 ∼ 2283 ∼= 2283 4689 ∼ 2202 ∼= 2202 4690 ∼ 2841 ∼= 2841 4691 ∼ 2372 ∼= 2372 4692 ∼ 2355 ∼= 2355 4693 ∼ 2372 ∼= 2372 4694 ∼ 2210 ∼= 2210 4695 ∼ 2274 ∼= 2274 4696 ∼ 2355 ∼= 2355 4697 ∼ 2274 ∼= 2274 4698 ∼ 2193 ∼= 2193 4699 ∼ 2844 ∼= 730 4700 ∼ 2358 ∼= 820 4701 ∼ 2426 ∼= 2277 4702 ∼ 2358 ∼= 820 4703 ∼ 2196 ∼= 802 4704 ∼ 2277 ∼= 2277 4705 ∼ 2426 ∼= 2277 4706 ∼ 2277 ∼= 2277 4707 ∼ 2264 ∼= 730 4708 ∼ 2838 ∼= 750 4709 ∼ 2352 ∼= 740 4710 ∼ 2399 ∼= 2399 4711 ∼ 2352 ∼= 740 4712 ∼ 2190 ∼= 750 4713 ∼ 2271 ∼= 2271 4714 ∼ 2399 ∼= 2399 4715 ∼ 2271 ∼= 2271 4716 ∼ 2237 ∼= 2237 4717 ∼ 2841 ∼= 2841 4718 ∼ 2355 ∼= 2355 4719 ∼ 2372 ∼= 2372 4720 ∼ 2355 ∼= 2355 4721 ∼ 2193 ∼= 2193 4722 ∼ 2274 ∼= 2274 4723 ∼ 2372 ∼= 2372 4724 ∼ 2274 ∼= 2274 4725 ∼ 2210 ∼= 2210 4726 ∼ 2838 ∼= 750 4727 ∼ 2352 ∼= 740 4728 ∼ 2399 ∼= 2399 4729 ∼ 2352 ∼= 740 4730 ∼ 2190 ∼= 750 4731 ∼ 2271 ∼= 2271 4732 ∼ 2399 ∼= 2399 4733 ∼ 2271 ∼= 2271 4734 ∼ 2237 ∼= 2237 4735 ∼ 1090 ∼= 1090 4736 ∼ 820 ∼= 820 4737 ∼ 820 ∼= 820 4738 ∼ 820 ∼= 820 4739 ∼ 730 ∼= 730 4740 ∼ 730 ∼= 730 4741 ∼ 820 ∼= 820 44 Classification of groups generated by automata 4742 ∼ 730 ∼= 730 4743 ∼ 730 ∼= 730 4744 ∼ 1090 ∼= 1090 4745 ∼ 820 ∼= 820 4746 ∼ 820 ∼= 820 4747 ∼ 820 ∼= 820 4748 ∼ 730 ∼= 730 4749 ∼ 730 ∼= 730 4750 ∼ 820 ∼= 820 4751 ∼ 730 ∼= 730 4752 ∼ 730 ∼= 730 4753 ∼ 2841 ∼= 2841 4754 ∼ 2355 ∼= 2355 4755 ∼ 2372 ∼= 2372 4756 ∼ 2355 ∼= 2355 4757 ∼ 2193 ∼= 2193 4758 ∼ 2274 ∼= 2274 4759 ∼ 2372 ∼= 2372 4760 ∼ 2274 ∼= 2274 4761 ∼ 2210 ∼= 2210 4762 ∼ 1090 ∼= 1090 4763 ∼ 820 ∼= 820 4764 ∼ 820 ∼= 820 4765 ∼ 820 ∼= 820 4766 ∼ 730 ∼= 730 4767 ∼ 730 ∼= 730 4768 ∼ 820 ∼= 820 4769 ∼ 730 ∼= 730 4770 ∼ 730 ∼= 730 4771 ∼ 1090 ∼= 1090 4772 ∼ 820 ∼= 820 4773 ∼ 820 ∼= 820 4774 ∼ 820 ∼= 820 4775 ∼ 730 ∼= 730 4776 ∼ 730 ∼= 730 4777 ∼ 820 ∼= 820 4778 ∼ 730 ∼= 730 4779 ∼ 730 ∼= 730 4780 ∼ 2853 ∼= 2853 4781 ∼ 2367 ∼= 2367 4782 ∼ 2423 ∼= 2423 4783 ∼ 2367 ∼= 2367 4784 ∼ 2205 ∼= 775 4785 ∼ 2286 ∼= 2286 4786 ∼ 2423 ∼= 2423 4787 ∼ 2286 ∼= 2286 4788 ∼ 2261 ∼= 2261 4789 ∼ 2851 ∼= 929 4790 ∼ 2365 ∼= 2365 4791 ∼ 2396 ∼= 2396 4792 ∼ 2365 ∼= 2365 4793 ∼ 2203 ∼= 2203 4794 ∼ 2284 ∼= 2284 4795 ∼ 2396 ∼= 2396 4796 ∼ 2284 ∼= 2284 4797 ∼ 2234 ∼= 2234 4798 ∼ 2852 ∼= 849 4799 ∼ 2366 ∼= 2366 4800 ∼ 2369 ∼= 2369 4801 ∼ 2366 ∼= 2366 4802 ∼ 2204 ∼= 2204 4803 ∼ 2285 ∼= 2285 4804 ∼ 2369 ∼= 2369 4805 ∼ 2285 ∼= 2285 4806 ∼ 2207 ∼= 2207 4807 ∼ 2847 ∼= 929 4808 ∼ 2361 ∼= 2361 4809 ∼ 2398 ∼= 2398 4810 ∼ 2361 ∼= 2361 4811 ∼ 2199 ∼= 2199 4812 ∼ 2280 ∼= 2280 4813 ∼ 2398 ∼= 2398 4814 ∼ 2280 ∼= 2280 4815 ∼ 2236 ∼= 2236 4816 ∼ 1090 ∼= 1090 4817 ∼ 820 ∼= 820 4818 ∼ 820 ∼= 820 4819 ∼ 820 ∼= 820 4820 ∼ 730 ∼= 730 4821 ∼ 730 ∼= 730 4822 ∼ 820 ∼= 820 4823 ∼ 730 ∼= 730 4824 ∼ 730 ∼= 730 4825 ∼ 1090 ∼= 1090 4826 ∼ 820 ∼= 820 4827 ∼ 820 ∼= 820 4828 ∼ 820 ∼= 820 4829 ∼ 730 ∼= 730 4830 ∼ 730 ∼= 730 4831 ∼ 820 ∼= 820 4832 ∼ 730 ∼= 730 4833 ∼ 730 ∼= 730 4834 ∼ 2850 ∼= 2850 4835 ∼ 2364 ∼= 2364 4836 ∼ 2371 ∼= 2371 4837 ∼ 2364 ∼= 2364 4838 ∼ 2202 ∼= 2202 4839 ∼ 2283 ∼= 2283 4840 ∼ 2371 ∼= 2371 4841 ∼ 2283 ∼= 2283 4842 ∼ 2209 ∼= 2209 4843 ∼ 1090 ∼= 1090 4844 ∼ 820 ∼= 820 4845 ∼ 820 ∼= 820 4846 ∼ 820 ∼= 820 4847 ∼ 730 ∼= 730 4848 ∼ 730 ∼= 730 4849 ∼ 820 ∼= 820 4850 ∼ 730 ∼= 730 4851 ∼ 730 ∼= 730 4852 ∼ 1090 ∼= 1090 4853 ∼ 820 ∼= 820 4854 ∼ 820 ∼= 820 4855 ∼ 820 ∼= 820 4856 ∼ 730 ∼= 730 4857 ∼ 730 ∼= 730 4858 ∼ 820 ∼= 820 4859 ∼ 730 ∼= 730 4860 ∼ 730 ∼= 730 4861 ∼ 2889 ∼= 750 4862 ∼ 2403 ∼= 2287 4863 ∼ 2424 ∼= 966 4864 ∼ 2403 ∼= 2287 4865 ∼ 2241 ∼= 739 4866 ∼ 2322 ∼= 2322 4867 ∼ 2424 ∼= 966 4868 ∼ 2322 ∼= 2322 4869 ∼ 2262 ∼= 750 4870 ∼ 2887 ∼= 731 4871 ∼ 2401 ∼= 2401 4872 ∼ 2401 ∼= 2401 4873 ∼ 2401 ∼= 2401 4874 ∼ 2239 ∼= 2239 4875 ∼ 2320 ∼= 2294 4876 ∼ 2401 ∼= 2401 4877 ∼ 2320 ∼= 2294 4878 ∼ 2239 ∼= 2239 4879 ∼ 2860 ∼= 2212 4880 ∼ 2402 ∼= 2402 4881 ∼ 2374 ∼= 821 4882 ∼ 2402 ∼= 2402 4883 ∼ 2240 ∼= 2240 4884 ∼ 2293 ∼= 2293 4885 ∼ 2374 ∼= 821 4886 ∼ 2293 ∼= 2293 4887 ∼ 2212 ∼= 2212 4888 ∼ 1091 ∼= 731 4889 ∼ 821 ∼= 821 4890 ∼ 821 ∼= 821 4891 ∼ 821 ∼= 821 4892 ∼ 731 ∼= 731 4893 ∼ 731 ∼= 731 4894 ∼ 821 ∼= 821 4895 ∼ 731 ∼= 731 4896 ∼ 731 ∼= 731 4897 ∼ 2874 ∼= 820 4898 ∼ 2395 ∼= 2395 4899 ∼ 2388 ∼= 821 4900 ∼ 2395 ∼= 2395 4901 ∼ 2233 ∼= 2233 4902 ∼ 2307 ∼= 2307 4903 ∼ 2388 ∼= 821 4904 ∼ 2307 ∼= 2307 4905 ∼ 2226 ∼= 820 4906 ∼ 2851 ∼= 929 4907 ∼ 2396 ∼= 2396 4908 ∼ 2365 ∼= 2365 4909 ∼ 2396 ∼= 2396 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 4910 ∼ 2234 ∼= 2234 4911 ∼ 2284 ∼= 2284 4912 ∼ 2365 ∼= 2365 4913 ∼ 2284 ∼= 2284 4914 ∼ 2203 ∼= 2203 4915 ∼ 1091 ∼= 731 4916 ∼ 821 ∼= 821 4917 ∼ 821 ∼= 821 4918 ∼ 821 ∼= 821 4919 ∼ 731 ∼= 731 4920 ∼ 731 ∼= 731 4921 ∼ 821 ∼= 821 4922 ∼ 731 ∼= 731 4923 ∼ 731 ∼= 731 4924 ∼ 2847 ∼= 929 4925 ∼ 2398 ∼= 2398 4926 ∼ 2361 ∼= 2361 4927 ∼ 2398 ∼= 2398 4928 ∼ 2236 ∼= 2236 4929 ∼ 2280 ∼= 2280 4930 ∼ 2361 ∼= 2361 4931 ∼ 2280 ∼= 2280 4932 ∼ 2199 ∼= 2199 4933 ∼ 2838 ∼= 750 4934 ∼ 2399 ∼= 2399 4935 ∼ 2352 ∼= 740 4936 ∼ 2399 ∼= 2399 4937 ∼ 2237 ∼= 2237 4938 ∼ 2271 ∼= 2271 4939 ∼ 2352 ∼= 740 4940 ∼ 2271 ∼= 2271 4941 ∼ 2190 ∼= 750 4942 ∼ 2853 ∼= 2853 4943 ∼ 2367 ∼= 2367 4944 ∼ 2423 ∼= 2423 4945 ∼ 2367 ∼= 2367 4946 ∼ 2205 ∼= 775 4947 ∼ 2286 ∼= 2286 4948 ∼ 2423 ∼= 2423 4949 ∼ 2286 ∼= 2286 4950 ∼ 2261 ∼= 2261 4951 ∼ 2847 ∼= 929 4952 ∼ 2361 ∼= 2361 4953 ∼ 2398 ∼= 2398 4954 ∼ 2361 ∼= 2361 4955 ∼ 2199 ∼= 2199 4956 ∼ 2280 ∼= 2280 4957 ∼ 2398 ∼= 2398 4958 ∼ 2280 ∼= 2280 4959 ∼ 2236 ∼= 2236 4960 ∼ 2850 ∼= 2850 4961 ∼ 2364 ∼= 2364 4962 ∼ 2371 ∼= 2371 4963 ∼ 2364 ∼= 2364 4964 ∼ 2202 ∼= 2202 4965 ∼ 2283 ∼= 2283 4966 ∼ 2371 ∼= 2371 4967 ∼ 2283 ∼= 2283 4968 ∼ 2209 ∼= 2209 4969 ∼ 2851 ∼= 929 4970 ∼ 2365 ∼= 2365 4971 ∼ 2396 ∼= 2396 4972 ∼ 2365 ∼= 2365 4973 ∼ 2203 ∼= 2203 4974 ∼ 2284 ∼= 2284 4975 ∼ 2396 ∼= 2396 4976 ∼ 2284 ∼= 2284 4977 ∼ 2234 ∼= 2234 4978 ∼ 1090 ∼= 1090 4979 ∼ 820 ∼= 820 4980 ∼ 820 ∼= 820 4981 ∼ 820 ∼= 820 4982 ∼ 730 ∼= 730 4983 ∼ 730 ∼= 730 4984 ∼ 820 ∼= 820 4985 ∼ 730 ∼= 730 4986 ∼ 730 ∼= 730 4987 ∼ 1090 ∼= 1090 4988 ∼ 820 ∼= 820 4989 ∼ 820 ∼= 820 4990 ∼ 820 ∼= 820 4991 ∼ 730 ∼= 730 4992 ∼ 730 ∼= 730 4993 ∼ 820 ∼= 820 4994 ∼ 730 ∼= 730 4995 ∼ 730 ∼= 730 4996 ∼ 2852 ∼= 849 4997 ∼ 2366 ∼= 2366 4998 ∼ 2369 ∼= 2369 4999 ∼ 2366 ∼= 2366 5000 ∼ 2204 ∼= 2204 5001 ∼ 2285 ∼= 2285 5002 ∼ 2369 ∼= 2369 5003 ∼ 2285 ∼= 2285 5004 ∼ 2207 ∼= 2207 5005 ∼ 1090 ∼= 1090 5006 ∼ 820 ∼= 820 5007 ∼ 820 ∼= 820 5008 ∼ 820 ∼= 820 5009 ∼ 730 ∼= 730 5010 ∼ 730 ∼= 730 5011 ∼ 820 ∼= 820 5012 ∼ 730 ∼= 730 5013 ∼ 730 ∼= 730 5014 ∼ 1090 ∼= 1090 5015 ∼ 820 ∼= 820 5016 ∼ 820 ∼= 820 5017 ∼ 820 ∼= 820 5018 ∼ 730 ∼= 730 5019 ∼ 730 ∼= 730 5020 ∼ 820 ∼= 820 5021 ∼ 730 ∼= 730 5022 ∼ 730 ∼= 730 5023 ∼ 2880 ∼= 730 5024 ∼ 2394 ∼= 820 5025 ∼ 2422 ∼= 820 5026 ∼ 2394 ∼= 820 5027 ∼ 2232 ∼= 730 5028 ∼ 2313 ∼= 2277 5029 ∼ 2422 ∼= 820 5030 ∼ 2313 ∼= 2277 5031 ∼ 2260 ∼= 802 5032 ∼ 2874 ∼= 820 5033 ∼ 2388 ∼= 821 5034 ∼ 2395 ∼= 2395 5035 ∼ 2388 ∼= 821 5036 ∼ 2226 ∼= 820 5037 ∼ 2307 ∼= 2307 5038 ∼ 2395 ∼= 2395 5039 ∼ 2307 ∼= 2307 5040 ∼ 2233 ∼= 2233 5041 ∼ 2854 ∼= 847 5042 ∼ 2391 ∼= 2391 5043 ∼ 2368 ∼= 739 5044 ∼ 2391 ∼= 2391 5045 ∼ 2229 ∼= 2229 5046 ∼ 2287 ∼= 2287 5047 ∼ 2368 ∼= 739 5048 ∼ 2287 ∼= 2287 5049 ∼ 2206 ∼= 748 5050 ∼ 2874 ∼= 820 5051 ∼ 2388 ∼= 821 5052 ∼ 2395 ∼= 2395 5053 ∼ 2388 ∼= 821 5054 ∼ 2226 ∼= 820 5055 ∼ 2307 ∼= 2307 5056 ∼ 2395 ∼= 2395 5057 ∼ 2307 ∼= 2307 5058 ∼ 2233 ∼= 2233 5059 ∼ 1090 ∼= 1090 5060 ∼ 820 ∼= 820 5061 ∼ 820 ∼= 820 5062 ∼ 820 ∼= 820 5063 ∼ 730 ∼= 730 5064 ∼ 730 ∼= 730 5065 ∼ 820 ∼= 820 5066 ∼ 730 ∼= 730 5067 ∼ 730 ∼= 730 5068 ∼ 1090 ∼= 1090 5069 ∼ 820 ∼= 820 5070 ∼ 820 ∼= 820 5071 ∼ 820 ∼= 820 5072 ∼ 730 ∼= 730 5073 ∼ 730 ∼= 730 5074 ∼ 820 ∼= 820 5075 ∼ 730 ∼= 730 5076 ∼ 730 ∼= 730 5077 ∼ 2854 ∼= 847 46 Classification of groups generated by automata 5078 ∼ 2391 ∼= 2391 5079 ∼ 2368 ∼= 739 5080 ∼ 2391 ∼= 2391 5081 ∼ 2229 ∼= 2229 5082 ∼ 2287 ∼= 2287 5083 ∼ 2368 ∼= 739 5084 ∼ 2287 ∼= 2287 5085 ∼ 2206 ∼= 748 5086 ∼ 1090 ∼= 1090 5087 ∼ 820 ∼= 820 5088 ∼ 820 ∼= 820 5089 ∼ 820 ∼= 820 5090 ∼ 730 ∼= 730 5091 ∼ 730 ∼= 730 5092 ∼ 820 ∼= 820 5093 ∼ 730 ∼= 730 5094 ∼ 730 ∼= 730 5095 ∼ 1090 ∼= 1090 5096 ∼ 820 ∼= 820 5097 ∼ 820 ∼= 820 5098 ∼ 820 ∼= 820 5099 ∼ 730 ∼= 730 5100 ∼ 730 ∼= 730 5101 ∼ 820 ∼= 820 5102 ∼ 730 ∼= 730 5103 ∼ 730 ∼= 730 5104 through 5832 ∼ 1090 ≃ 1090. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 8. Group information We use the following notation: • Rels - a list of some relators in the group. In most cases these are the first few relators in the length-lexicographic order, but in some cases (more precisely, for the automata numbered by 744, 753, 776, 840, 843, 858, 885, 888, 956, 965, 2209, 2210, 2213, 2234, 2261, 2274, 2293, 2355, 2364, 2396, 2402, 2423) there could be some shorter relators. In most cases the given list does not give a presentation of the group (exception are the finite and abelian groups, and the automata numbered by 820, 846, 870, 2212, 2240, 2294). • SF - these numbers represent the size of the factors G/ StabG(n), for n ≥ 0. • Gr - these numbers represent the first few values of the growth function γG(n), for n ≥ 0, with respect to the generating system a, b, c (γG(n) counts the number of elements of length at most n in G). Automaton number 1 a = (a, a) b = (a, a) c = (a, a) Group: Trivial Group Contracting: yes Self-replicating: yes Rels: a, b, c SF: 20,20,20,20,20,20,20,20,20 Gr: 1,1,1,1,1,1,1,1,1,1,1 a b c 1 1 1 0,1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 51 102 154 205 256 307 358 410 461 512 48 Classification of groups generated by automata Automaton number 730 a = σ(a, a) b = (a, a) c = (a, a) Group: Klein Group Contracting: yes Self-replicating: no Rels: b−1c, a2, b2, abab SF: 20,21,22,22,22,22,22,22,22 Gr: 1,3,4,4,4,4,4,4,4,4,4 a b c σ 1 1 0,1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 12.8 25.6 38.4 51.2 64.0 76.8 89.6 102.4 115.2 128.0 Automaton number 731 a = σ(b, a) b = (a, a) c = (a, a) Group: Z Contracting: yes Self-replicating: yes Rels: b−1c, ba2 SF: 20,21,22,23,24,25,26,27,28 Gr: 1,5,9,13,17,21,25,29,33,37,41 a b c σ 1 1 0 1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 18.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 739 a = σ(a, a) b = (b, a) c = (a, a) Group: C2 ⋉ ( Z ≀ C2 ) Contracting: yes Self-replicating: no Rels: a2, b2, c2, (ac)2, (acbab)2 SF: 20,21,23,26,28,210,212,214,216 Gr: 1,4,9,17,30,47,68,93,122,155,192 a b c σ 1 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.6 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 16.0 Automaton number 740 a = σ(b, a) b = (b, a) c = (a, a) Group: Contracting: no Self-replicating: no Rels: (a−1b)2, (b−1c)2, a−1c−1ac−1b2, [a, b]2 SF: 20,21,23,26,29,211,214,216,218 Gr: 1,7,33,135,495,1725 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 50 Classification of groups generated by automata Automaton number 741 a = σ(c, a) b = (b, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: ca2, b−1a−3b−1ababa, b−1a−6b−1a−2ba−2ba−2 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,29,115,441,1643 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.7 3.4 5.1 6.8 8.5 10.2 11.9 13.6 15.3 17.0 Automaton number 744 a = σ(c, b) b = (b, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: [a2ca−1bc−1b−1a−1, aca−1bc−1b−1], abcb−1ac−1a−2bcb−1ab−1aca−1bc−1a−1bc−1b−1, abcb−1ab−1a−2bcb−1ac−1aba−1bc−1b−1ca−1bc−1b−1, abcb−1ab−1a−2bcb−1ab−1a· ba−1bc−1a−1bc−1b−1 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 748 a = σ(a, a) b = (c, a) c = (a, a) Group: D4 × C2 Contracting: yes Self-replicating: no Rels: a2, b2, c2, acac, bcbc, abababab SF: 20,21,23,24,24,24,24,24,24 Gr: 1,4,8,12,15,16,16,16,16,16,16 a b c σ 1 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 6.4 12.8 19.2 25.6 32.0 38.4 44.8 51.2 57.6 64.0 Automaton number 749 a = σ(b, a) b = (c, a) c = (a, a) Group: Contracting: n/a Self-replicating: yes Rels: a−1c−1bab−1a−1cb−1ab, a−1c−1bac−1a−1cb−1ac SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4667 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 52 Classification of groups generated by automata Automaton number 750 a = σ(c, a) b = (c, a) c = (a, a) Group: C2 ≀ Z Contracting: yes Self-replicating: no Rels: ca2, (a−1b)2, [b, c] SF: 20,21,23,25,27,29,211,213,215 Gr: 1,7,23,49,87,137,199,273,359 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.4 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 34.0 Automaton number 752 a = σ(b, b) b = (c, a) c = (a, a) Group: virtually Z 3 Contracting: yes Self-replicating: no Rels: a2, b2, c2, (acbab)2, (acacb)2, (abc)2(acb)2, acbcbabacbcbab, abcbacbabcbacb, acbcacbacbcacb, acacbcbacacbcb, abc(bca)2cbcbacb, a(cb)3aba(cb)3ab, abcbcbacbabcbcbacb, acbcbcacbacbcbcacb SF: 20,21,23,25,27,28,210,211,213 Gr: 1,4,10,22,46,84,140,217,319,448 a b c σ 1 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.6 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 16.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 753 a = σ(c, b) b = (c, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: aba−1b−1ab−1ca−1ba−1b−1ab−1cac−1b· a−1bab−1a−1c−1ba−1bab−1, aba−1b−1ab−1ca−1c−1ba−1c−1bab−1ca· c−1ba−1bab−1a−1c−1ba−1b−1cab−1c SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 771 a = σ(c, b) b = (b, b) c = (a, a) Group: Z 2 Contracting: yes Self-replicating: yes Rels: b, a−1c−1ac SF: 20,21,22,23,24,25,26,27,28 Gr: 1,5,13,25,41,61,85,113,145,181,221 Limit space: 2-dimensional torus T2 a b c σ 1 1 0 1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 32.0 54 Classification of groups generated by automata Automaton number 775 a = σ(a, a) b = (c, b) c = (a, a) Group: C2 ⋉ IMG ( ( z−1 z+1 )2 ) Contracting: yes Self-replicating: yes Rels: a2, b2, c2, acac, acbcbabcbcabcbabcb SF: 20,21,22,24,26,29,215,226,248 Gr: 1,4,9,17,30,51,85,140,229,367,579 Limit space: a b c σ 1 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.00 4.40 8.80 13.20 17.60 22.00 26.40 30.80 35.20 39.60 44.00 Automaton number 776 a = σ(b, a) b = (c, b) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: aba−1b−1a2c−1ab−1a−1bcb−1ac−1a−1ba−1· b−1a2c−1ab−1a−1bcb−1ac−1aca−1bc−1b−1ab· a−1ca−2bab−1a−1ca−1bc−1b−1aba−1ca−2bab−1, aba−1b−1a2c−1ab−1a−1bcb−1ac−1a−1cba−1· b−1a2c−1ab−1a−1bc−1b−1aba−1ca−2· bab−1aca−1bc−1b−1aba−1ca−2bab−1· a−1ba−1b−1a2c−1ab−1a−1bcb−1· aba−1ca−2bab−1c−1 SF: 20,21,22,24,27,213,224,246,289 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 777 a = σ(c, a) b = (c, b) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: ca2, b−1a5b−1a−1ba−3ba−1 SF: 20,21,22,24,27,213,224,246,289 Gr: 1,7,29,115,441,1695 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.4 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 14.0 Automaton number 779 a = σ(b, b) b = (c, b) c = (a, a) Group: Contracting: yes Self-replicating: yes Rels: a2, b2, c2, acabcabcbabacabcabcbab, acbcbacacabcbcabcbabcb SF: 20,21,22,24,26,29,215,226,248 Gr: 1,4,10,22,46,94,190,382,766,1534,3070,6120 a b c σ 1 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 56 Classification of groups generated by automata Automaton number 780 a = σ(c, b) b = (c, b) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, [ba−1, c] SF: 20,21,22,24,26,29,215,227,249 Gr: 1,7,35,159,705,3107 a b c σ 1 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Automaton number 802 a = σ(a, a) b = (c, c) c = (a, a) Group: C2 × C2 × C2 Contracting: yes Self-replicating: no Rels: a2, b2, c2, [a, b], [a, c], [b, c] SF: 20,21,22,23,23,23,23,23,23 Gr: 1,4,7,8,8,8,8,8,8,8,8 a b c σ 1 1 0,1 0,10,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 19.2 38.4 57.6 76.8 96.0 115.2 134.4 153.6 172.8 192.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 820 a = σ(a, a) b = (b, a) c = (b, a) Group: D∞ Contracting: yes Self-replicating: yes Rels: b−1c, a2, b2 SF: 20,21,23,24,25,26,27,28,29 Gr: 1,3,5,7,9,11,13,15,17,19,21 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.6 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 16.0 Automaton number 821 a = σ(b, a) b = (b, a) c = (b, a) Group: Lamplighter group Z ≀ C2 Contracting: no Self-replicating: yes Rels: b−1c, (a−1b)2, [a, b]2, a−3baba−2b−1a2b SF: 20,21,23,25,26,28,29,210,211 Gr: 1,5,15,39,92,208,452,964,2016 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 5 10 15 20 25 30 35 40 45 50 58 Classification of groups generated by automata Automaton number 838 a = σ(a, a) b = (a, b) c = (b, a) Group: C2 ⋉ 〈s, t ∣ ∣ s2 = t2〉 Contracting: yes Self-replicating: no Rels: a2, b2, c2, abcacb SF: 20,21,23,25,27,29,211,213,215 Gr: 1,4,10,19,31,46,64,85,109,136 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 13.4 26.8 40.2 53.6 67.0 80.4 93.8 107.2 120.6 134.0 Automaton number 840 a = σ(c, a) b = (a, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: abac−1a−2bac−1aca−1b−1ca−1b−1, abac−1a−2cac−1b−1caca−1b−1c−1bca−1c−1, acac−1b−1ca−2bac−1ac−1bca−2b−1, acac−1b−1ca−2cac−1b−1cac−1bca−1c−2bca−1c−1 SF: 20,21,23,25,27,210,215,225,241 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 843 a = σ(c, b) b = (a, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: acab−1a−2cab−1aba−1c−1ba−1c−1, acab−1a−2cb−1ab−1caba−1c−2ba−1bc−1, acb−1ab−1ca−2cab−1ac−1ba−1bc−1ba−1c−1, acb−1ab−1ca−2cb−1ab−1cac−1ba−1bc−2ba−1bc−1 SF: 20,21,23,25,28,214,224,243,281 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 846 a = σ(c, c) b = (a, b) c = (b, a) Group: C2 ∗ C2 ∗ C2 Contracting: no Self-replicating: no Rels: a2, b2, c2 SF: 20,21,23,25,27,210,213,216,219 Gr: 1,4,10,22,46,94,190,382,766,1534 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 2.7 5.4 8.1 10.8 13.5 16.2 18.9 21.6 24.3 27.0 60 Classification of groups generated by automata Automaton number 847 a = σ(a, a) b = (b, b) c = (b, a) Group: D4 Contracting: yes Self-replicating: no Rels: b, a2, c2, acacacac SF: 20,21,23,23,23,23,23,23,23 Gr: 1,3,5,7,8,8,8,8,8,8,8 a b c σ 1 1 0,1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 12.8 25.6 38.4 51.2 64.0 76.8 89.6 102.4 115.2 128.0 Automaton number 849 a = σ(c, a) b = (b, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: b, [ac−1a−1, c], [a2, c−1] · [c, a−2], [a−1, c−2] · [a−1, c2] SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,5,17,53,153,421,1125,2945,7589 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.4 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 34.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 852 a = σ(c, b) b = (b, b) c = (b, a) Group: IMG(z2 − 1) Contracting: yes Self-replicating: yes Rels: b, [ac−1a−1, c], [c, a2] · [c, a−2], [a−1, c−2] · [a−1, c2] SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,5,17,53,153,421,1125,2945,7545 Limit space: a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.00 4.30 8.60 12.90 17.20 21.50 25.80 30.10 34.40 38.70 43.00 Automaton number 856 a = σ(a, a) b = (c, b) c = (b, a) Group: C2 ⋉G2850 Contracting: no Self-replicating: yes Rels: a2, b2, c2, acbcacbcabcacacacb SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,190,382,766, 1525,3025,5998,11890,23532 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 62 Classification of groups generated by automata Automaton number 857 a = σ(b, a) b = (c, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (a−1c)2, (a−1b)4, (a−1b−1ac)2, (b−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,758,3460 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 858 a = σ(c, a) b = (c, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: abca−1c−1ab−1a2c−1b−1a−1bca−1c−1a· b−1a2c−1b−1abca−2ba−1cac−1b−1a−1· bca−2ba−1cac−1b−1, abca−1c−1ab−1a2c−1b−1a−1cba−1b−1ab−1a· bca−2ba−1cac−1b−1a−1ba−1bab−1c−1 SF: 20,21,23,27,213,224,246,290,2176 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.4 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 14.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 860 a = σ(b, b) b = (c, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: a2, b2, c2, acbacacabcabab SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,190,375,731,1422,2762,5350 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 2.2 4.4 6.6 8.8 11.0 13.2 15.4 17.6 19.8 22.0 Automaton number 861 a = σ(c, b) b = (c, b) c = (b, a) Group: Contracting: n/a Self-replicating: yes Rels: (a−1b)2, (b−1c)2, [a, b]2, [b, c]2 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,599,2485 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 6.9 13.8 20.7 27.6 34.5 41.4 48.3 55.2 62.1 69.0 64 Classification of groups generated by automata Automaton number 864 a = σ(c, c) b = (c, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: a2, b2, c2, abcabcbabcbacbabab SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,190,382,766,1525, 3025,5998,11890 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 Automaton number 866 a = σ(b, a) b = (a, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (ca−1)2, ba−2cab−1ab−1c−1aba−1, cab−1cb−1a−1cbc−1ba−2 SF: 20,21,23,25,29,215,226,248,292 Gr: 1,7,35,165,769,3575 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 869 a = σ(b, b) b = (a, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: a2, b2, c2, acbcacbcabcacacacb SF: 20,21,23,24,26,29,215,226,248 Gr: 1,4,10,22,46,94,190,382,766,1525 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 870 a = σ(c, b) b = (a, c) c = (b, a) Group: BS(1, 3) Contracting: no Self-replicating: yes Rels: a−1ca−1b, (b−1a)b−1 (b−1a)−3 SF: 20,21,23,24,26,28,210,212,214 Gr: 1,7,33,127,433,1415 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 32.0 66 Classification of groups generated by automata Automaton number 874 a = σ(a, a) b = (b, c) c = (b, a) Group: C2 ⋉G2852 Contracting: no Self-replicating: yes Rels: a2, b2, c2, abcabcacbacb, abcbcabcacbcbacb SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,184,352,664,1244,2320,4288 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.2 18.0 Automaton number 875 a = σ(b, a) b = (b, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (a−1c)2, (b−1c)2, (a−1b)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,607,2563 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 876 a = σ(c, a) b = (b, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: a−2bcb−2a2c−1b, a−2cb−1a2c−2bc SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,937,4667 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 Automaton number 878 a = σ(b, b) b = (b, c) c = (b, a) Group: C2 ⋉ IMG(1 − 1 z2 ) Contracting: yes Self-replicating: yes Rels: a2, b2, c2, abcabcacbacb, abcbcabcacbcbacb SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,184,352,664,1244,2296,4198,7612 Limit space: a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 2.4 4.8 7.2 9.6 12.0 14.4 16.8 19.2 21.6 24.0 68 Classification of groups generated by automata Automaton number 879 a = σ(c, b) b = (b, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, a−1ca−1cb−1ac−1ac−1b SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,769,3567 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.4 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 14.0 Automaton number 882 a = σ(c, c) b = (b, c) c = (b, a) Group: Contracting: n/a Self-replicating: yes Rels: a2, b2, c2, abcabcacbacb, abcbcabcacbcbacb SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,184,352,664,1244 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.6 3.2 4.8 6.4 8.0 9.6 11.2 12.8 14.4 16.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 883 a = σ(a, a) b = (c, c) c = (b, a) Group: C2 ⋉G2841 Contracting: no Self-replicating: yes Rels: a2, b2, c2, acbcbacbcacbcabcbcabab, acbacbcacabacbacbcacab SF: 20,21,23,26,29,214,224,243,280 Gr: 1,4,10,22,46,94,190,382,766,1534,3070,6120 a b c σ 1 1 0,1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 Automaton number 884 a = σ(b, a) b = (c, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (a−1c)2, (b−1c)2, [b, ac−1] SF: 20,21,23,26,29,215,227,249,293 Gr: 1,7,33,135,529,2051 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 70 Classification of groups generated by automata Automaton number 885 a = σ(c, a) b = (c, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: acba−1b−1ac−1a−1cba−1b−1ac−1aca−1· bab−1c−1a−1ca−1bab−1c−1, acba−1b−1ac−1a−1ca−1c−1b−1a3c−1aca−1b· ab−1c−1a−1ca−3bcac−1 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 887 a = σ(b, b) b = (c, c) c = (b, a) Group: Contracting: n/a Self-replicating: yes Rels: a2, b2, c2, babacbcbacbcacbcabcbca, bacacbcabcabacacbcabca SF: 20,21,23,26,29,214,224,243,280 Gr: 1,4,10,22,46,94,190,382,766,1534,3070,6120 a b c σ 1 1 0,1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 888 a = σ(c, b) b = (c, c) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: aca−1ba−2ca−1bab−1ac−1b−1ac−1, aca−1ba−3bab−1a2b−1ac−1a−1ba−1b−1a, bab−1a−1ca−1b2a−1b−1ab−1ac−1, bab−1a−2bab−1aba−2b−1a SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 Automaton number 891 a = σ(c, c) b = (c, c) c = (b, a) Group: C2 ⋉ Lampighter Contracting: no Self-replicating: yes Rels: a2, b2, c2, abab, (acb)4, [acaca, bcacb], [acaca, bcbcb] SF: 20,21,23,26,27,29,210,211,212 Gr: 1,4,9,17,30,51,82,128,198,304 a b c σ 1 1 0,1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 7.2 14.4 21.6 28.8 36.0 43.2 50.4 57.6 64.8 72.0 72 Classification of groups generated by automata Automaton number 920 a = σ(b, a) b = (a, b) c = (c, a) Group: Contracting: n/a Self-replicating: yes Rels: (a−1b)2, [a, b]2, (a−1c−1ab)2 SF: 20,21,23,25,29,215,226,248,292 Gr: 1,7,35,165,757,3447 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Automaton number 923 a = σ(b, b) b = (a, b) c = (c, a) Group: Contracting: yes Self-replicating: yes Rels: a2, b2, c2, abcabcbabcbacbabab SF: 20,21,23,24,26,29,215,226,248 Gr: 1,4,10,22,46,94,190,382,766, 1525,3025,5998,11890 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 929 a = σ(b, a) b = (b, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: b, a−3cac−1ac−1ac SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,5,17,53,161,475,1387 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.9 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 19.0 Automaton number 937 a = σ(a, a) b = (c, b) c = (c, a) Group: C2 ⋉G929 Contracting: no Self-replicating: yes Rels: a2, b2, c2, abcabcacbacb, abcbcabcacbcbacb SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,184,352,664,1244 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 74 Classification of groups generated by automata Automaton number 938 a = σ(b, a) b = (c, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: a−2bcb−2a2c−1b, a−2cb−1a2c−2bc SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,937,4667 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 939 a = σ(c, a) b = (c, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (a−1c)2, (a−2cb)2, [a, c]2, [ca−1, ba−1b], a−1b−1abc−1a−1bca−1b SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,757,3427 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 941 a = σ(b, b) b = (c, b) c = (c, a) Group: C2 ⋉ IMG(z2 − 1) Contracting: yes Self-replicating: yes Rels: a2, b2, c2, abcabcacbacb, abcbcabcacbcbacb SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,184,352,664,1244 Limit space: a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 Automaton number 942 a = σ(c, b) b = (c, b) c = (c, a) Group: Contains Lamplighter group Contracting: no Self-replicating: yes Rels: (a−1b)2, (b−1c)2, [a, b]2, [b, c]2, (a−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,597,2465 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 7.7 15.4 23.1 30.8 38.5 46.2 53.9 61.6 69.3 77.0 76 Classification of groups generated by automata Automaton number 956 a = σ(b, a) b = (b, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: acba−1b−1ab−1a−1cba−1b−1ab−1aba−1· bab−1c−1a−1ba−1bab−1c−1, acba−1b−1ab−1a−1b−1ca−1caba−1bab−1c−1· a−2bc−1baba−1bab−1c−1a−1b−1cb−1a2cb· a−1b−1ab−1a−1c−1ac−1b SF: 20,21,23,27,213,224,246,290,2176 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 957 a = σ(c, a) b = (b, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (a−1c)2, (b−1c)2, [a, c]2, [b, c]2, (a−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,599,2485 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 959 a = σ(b, b) b = (b, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: a2, b2, c2, abcabcbabcbacbabab SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,190,382,766,1525 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 960 a = σ(c, b) b = (b, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, (a−2bc)2, (a−1c)4, (b−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,758,3460 a b c σ 1 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.1 2.2 3.3 4.4 5.5 6.6 7.7 8.8 9.9 11.0 78 Classification of groups generated by automata Automaton number 963 a = σ(c, c) b = (b, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: a2, b2, c2, acbacacabcabab SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,4,10,22,46,94,190,375,731, 1422,2762,5350,10322 a b c σ 1 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 Automaton number 965 a = σ(b, a) b = (c, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: acb−1a−1cb−1abc−1a−1bc−1, acb−1a−1cac−1b−1cbc−2bca−1c−1, acac−1b−1ca−2cb−1a2c−1bca−1c−1a−1bc−1, acac−1b−1ca−2cac−1b−1cac−1bca−1c−2bca−1c−1 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 966 a = σ(c, a) b = (c, c) c = (c, a) Group: Contracting: no Self-replicating: no Rels: (a−1c)2, (b−1c)2, [ca−1, b], [a, b]2, (a−2b2)2, (a−1b)4, [[c−1, a−1], cb−1] SF: 20,21,23,26,29,211,214,216,218 Gr: 1,7,33,135,495,1725 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 Automaton number 968 a = σ(b, b) b = (c, c) c = (c, a) Group: Virtually Z 5 Contracting: yes Self-replicating: no Rels: a2, b2, c2, (abc)2(acb)2, (cbcbaba)2, (cacbcba)2, (cabacbaba)2, ((ac)4b)2 SF: 20,21,23,26,29,213,217,221,225 Gr: 1,4,10,22,46,94,184,338,600,1022 a b c σ 1 1 0,1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 80 Classification of groups generated by automata Automaton number 969 a = σ(c, b) b = (c, c) c = (c, a) Group: Contracting: n/a Self-replicating: yes Rels: a−1c−1bab−1a−1cb−1ab, a−1c−1bac−1a−1cb−1ac SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4667 a b c σ 1 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 3 6 9 12 15 18 21 24 27 30 Automaton number 1090 a = σ(a, a) b = (b, b) c = (b, b) Group: C2 Contracting: yes Self-replicating: no Rels: b, c, a2 SF: 20,21,21,21,21,21,21,21,21 Gr: 1,2,2,2,2,2,2,2,2,2,2 a b c σ 1 1 0,1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 25.6 51.2 76.8 102.4 128.0 153.6 179.2 204.8 230.4 256.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2193 a = σ(c, b) b = σ(a, a) c = (a, a) Group: Contains Lamplighter group Contracting: no Self-replicating: yes Rels: [b, c], b2c2, a4, b4, (a2b)2, (abc)2, (a2c)2 SF: 20,21,23,26,27,29,210,211,212 Gr: 1,7,27,65,120,204,328, 512,792,1216 a b c σ σ 1 0 1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 19.2 38.4 57.6 76.8 96.0 115.2 134.4 153.6 172.8 192.0 Automaton number 2199 a = σ(c, a) b = σ(b, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: ca2, [a−1b, ab−1] SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,29,115,417,1505 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 82 Classification of groups generated by automata Automaton number 2202 a = σ(c, b) b = σ(b, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: cab2a SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,177,833,3909 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Automaton number 2203 a = σ(a, c) b = σ(b, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: ca2, [c−2ab, bc−2a] SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,29,115,441,1695 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2204 a = σ(b, c) b = σ(b, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: bcba2, [b−1a, ba−1], a−1ba2ba−2b−2aba2b−1a−2 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,177,825,3781 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 Automaton number 2207 a = σ(b, a) b = σ(c, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1] SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,929,4599 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 84 Classification of groups generated by automata Automaton number 2209 a = σ(a, b) b = σ(c, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: aca−2c−1acac−1a−2cac−1, aca−2b−1a−1cacac−1a−2c−1abac−1, aca−1b−1a−1c2a−1c−1ac−1abac−1a−2cac−1, aca−1b−1a−1c2a−1b−1a−1cac−1· abac−1a−2c−1abac−1, bca−1c−1ab−1ca−1c−1aba−1ca· c−1b−1a−1cac−1 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 Automaton number 2210 a = σ(b, b) b = σ(c, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: acbc−1b−1a−1cbc−1b−1abcb−1c−1a−1bcb−1c−1, bcbc−1b−2cbc−1bcb−2c−1, bcbc−1b−2ca−1b−1cabcb−1c−1a−1c−1bac−1, bca−1b−1cab−2cbc−1ba−1c−1bab−1c−1, bca−1b−1cab−2ca−1b−1caba−1c−1· bac−1a−1c−1bac−1 SF: 20,21,23,25,28,213,223,242,279 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2212 a = σ(a, c) b = σ(c, a) c = (a, a) Group: Klein bottle group Contracting: yes Self-replicating: no Rels: ca2, cb2 SF: 20,21,22,24,26,28,210,212,214 Gr: 1,7,19,37,61,91,127,169,217,271,331 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.9 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 19.0 Automaton number 2213 a = σ(b, c) b = σ(c, a) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: bcbc−1b−2cbc−1bcb−2c−1, acbc−1b−1a−1cbc−1b−1abcb−1c−1· a−1bcb−1c−1, acbc−1b−1a−1ba−1c−1b2c−1abcb−1c−1a−1· cb−2cab−1, aba−1c−1b2c−1a−1cbc−1b−1· acb−2cab−1a−1bcb−1c−1, SF: 20,21,22,23,25,28,214,225,247 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 86 Classification of groups generated by automata Automaton number 2229 a = σ(c, b) b = σ(b, b) c = (a, a) Group: C4 ⋉ Z 2 Contracting: yes Self-replicating: no Rels: b2, (ab)2, (bc)2, a4, c4, [a, c]2, (a−1c)4, (ac)4 SF: 20,21,23,26,29,211,213,215,217 Gr: 1,6,20,54,128,270,510,886,1452 a b c σ σ 1 0 1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 Automaton number 2233 a = σ(a, a) b = σ(c, b) c = (a, a) Group: Contracting: yes Self-replicating: yes Rels: a2, c2, abab, acac, cb2acbcbcab2cabcba SF: 20,21,23,26,29,215,226,248,291 Gr: 1,5,14,32,68,140,284,565,1106 a b c σ σ 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.3 2.6 3.9 5.2 6.5 7.8 9.1 10.4 11.7 13.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2234 a = σ(b, a) b = σ(c, b) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: ac−1a2c−1ab−1a−1c−1a2c−1ab−1ab· a−1ca−2ca−1ba−1ca−2c, ac−1a2c−1ab−1a−1cbac−1ab−1a−1c−1aba−1· ca−1b−1aba−1ca−2ca−1bac−1ab−1a−1ca· ba−1ca−1b−1c−1 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 2236 a = σ(a, b) b = σ(c, b) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1], a−1c−1acb−1ac−1a−1cb, a−1cac−1b−1aca−1c−1b SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,929,4579 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 88 Classification of groups generated by automata Automaton number 2237 a = σ(b, b) b = σ(c, b) c = (a, a) Group: Contracting: no Self-replicating: no Rels: [b−1a, ba−1], [c−1a, ca−1] SF: 20,21,23,26,29,215,226,245,281 Gr: 1,7,37,187,921,4511 a b c σ σ 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 2239 a = σ(a, c) b = σ(c, b) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: ca2, [ca−2cba−1, a−1ca−2cb] SF: 20,21,22,23,25,28,214,225,247 Gr: 1,7,29,115,441,1695 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.4 2.8 4.2 5.6 7.0 8.4 9.8 11.2 12.6 14.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2240 a = σ(b, c) b = σ(c, b) c = (a, a) Group: F3 Contracting: no Self-replicating: no Rels: SF: 20,21,22,24,27,210,214,221,234 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.00 4.40 8.80 13.20 17.60 22.00 26.40 30.80 35.20 39.60 44.00 Automaton number 2261 a = σ(b, a) b = σ(c, c) c = (a, a) Group: Contracting: no Self-replicating: yes Rels: acac−1a−2cac−1aca−2c−1, acac−1a−2cba−1c−1aca−1cb−1aca−1c−1· bc−1ac−1a−1cab−1c−1, bcac−1a−1b−1cac−1a−1baca−1c−1b−1aca−1c−1 SF: 20,21,22,24,26,29,215,226,248 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0,10,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 90 Classification of groups generated by automata Automaton number 2265 a = σ(c, b) b = σ(c, c) c = (a, a) Group: Contracting: no Self-replicating: no Rels: [b−1a, ba−1], a−1ca−1cb−1ac−1ac−1b, a−1cb−1cb−1ac−1bc−1b SF: 20,21,23,26,29,214,222,236,263 Gr: 1,7,37,187,929,4579,22521 a b c σ σ 1 0 1 0,10,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 2271 a = σ(c, a) b = σ(a, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1], a−1c2a−1b−1a2c−2b, a−1c2b−2abc−2b SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,929,4583 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2274 a = σ(c, b) b = σ(a, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: ac3b−1c−2b3c−3a−1c3b−1c−2b3c−3ac3b−3· c2bc−3a−1c3b−3c2bc−3, ac3b−1c−2b3c−3a−1c2ab−2c−1b3c−3ac3b−3· c2bc−3a−1c3b−3cb2a−1c−2, bc3b−1c−2b3c−3b−1c3b−1c−2b3c−3· bc3b−3c2bc−3b−1c3b−3c2bc−3 SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 2277 a = σ(c, c) b = σ(a, a) c = (b, a) Group: C2 ⋉ (Z × Z) Contracting: yes Self-replicating: yes Rels: a2, b2, c2, (acb)2 SF: 20,21,22,24,25,26,27,28,29 Gr: 1,4,10,19,31,46,64,85,109,136,166 Limit space: 2-dimensional sphere S2 a b c σ σ 1 0,1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 32.0 92 Classification of groups generated by automata Automaton number 2280 a = σ(c, a) b = σ(b, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, (b−1c)2, [a, b]2, [b, c]2, (a−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,597,2465 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 2283 a = σ(c, b) b = σ(b, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, (b−1c)2, [b, c]2 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,604,2534 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2284 a = σ(a, c) b = σ(b, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (b−1c)2, (a−1b)4, (bc−2a)2, (a−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,758,3460 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 2285 a = σ(b, c) b = σ(b, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (b−1c)2, [b−1a, ba−1], [(c−1a)2, c−1b], [(ca−1)2, cb−1] SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,761,3479 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 94 Classification of groups generated by automata Automaton number 2286 a = σ(c, c) b = σ(b, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: (b−1c)2, [a, bc−1] SF: 20,21,22,23,25,29,215,227,249 Gr: 1,7,35,159,705,3107 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Automaton number 2287 a = σ(a, a) b = σ(c, a) c = (b, a) Group: IMG ( z2+2 1−z2 ) Contracting: yes Self-replicating: yes Rels: a2, [a, b2], (b−1ac)2, [ba, c2], [c2, aca] SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,6,26,100,362,1246 Limit space: a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2293 a = σ(a, c) b = σ(c, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: cb−1a−1ca−1cb−1a−1cac−1abc−1a−1c−1abc−1a, cb−1a−1c2a−1c2b−1a−1c2b−1a−1ca−2c−1a· b2c−2ab−1a−1ca2c−1abc−2abc−2ac−1, ba−1cb−1a−1cab−1a−1cb−1a−1c· aba−1c−1abc−1ab−1a−1c−1abc−1a SF: 20,21,22,24,28,213,223,241,276 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 Automaton number 2294 a = σ(b, c) b = σ(c, a) c = (b, a) Group: BS(1,−3) Contracting: no Self-replicating: yes Rels: b−1ca−1c, (ca−1)a(ca−1)3 SF: 20,21,22,24,26,28,210,212,214 Gr: 1,7,33,127,433,1415 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.2 6.4 9.6 12.8 16.0 19.2 22.4 25.6 28.8 32.0 96 Classification of groups generated by automata Automaton number 2295 a = σ(c, c) b = σ(c, a) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1] SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,929,4599 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 Automaton number 2307 a = σ(c, a) b = σ(b, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: b2, a−2c−1bca2c−1bc, a−1c−1bc−2bcac2, a−1cbc−2bc−1ac2 SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,6,26,106,426,1681 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2322 a = σ(c, c) b = σ(c, b) c = (b, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1] SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,929,4599 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 2355 a = σ(c, b) b = σ(a, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: bca−2c−1bcac−1b−2cac−1, aca−1c−1ba−1ca−1c−1bab−1cac−1a−1b−1cac−1, abac−1bc−1b−1a−1ca−1c−1bab· cb−1ca−1b−1a−1b−1cac−1, aca−1c−1ba−1bac−1bc−1b−1a· b−1cac−1a−1bcb−1ca−1b−1 SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 98 Classification of groups generated by automata Automaton number 2361 a = σ(c, a) b = σ(b, a) c = (c, a) Group: Contracting: n/a Self-replicating: yes Rels: (a−1c)2, [b−1a, ba−1], [a, c]2, (b−1a−1c2)2, [ac−1, bc−1ba−1] SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,749,3343 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 2364 a = σ(c, b) b = σ(b, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: aca−1cb−1a−1ca−1cb−1abc−1ac−1a−1bc−1ac−1, bca−1cb−2ca−2ca−1b3c−1ac−1b−2ac−1a2c−1, bca−2ca−1ca−2ca−1bac−1a2c−1b−2ac−1a2c−1, bca−2ca−1ca−1cb−1ac−1a2c−2ac−1, bca−1cb−2ca−1cbc−1ac−2ac−1 SF: 20,21,23,26,212,224,246,290,2176 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0 1 2 3 4 5 6 7 8 9 10 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2365 a = σ(a, c) b = σ(b, a) c = (c, a) Group: Contracting: n/a Self-replicating: yes Rels: (a−1b)2, (a−1c)2, [a, c]2 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,604,2534 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 2366 a = σ(b, c) b = σ(b, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1], a−1c−1acb−1ac−1a−1cb, a−1cbc−1b−1acb−1c−1b SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,37,187,929,4579 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 100 Classification of groups generated by automata Automaton number 2367 a = σ(c, c) b = σ(b, a) c = (c, a) Group: Contracting: yes Self-replicating: yes Rels: a2, c2, b−2cacb2cac SF: 20,21,23,25,28,214,225,247,290 Gr: 1,5,17,53,161,480,1422 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 Automaton number 2369 a = σ(b, a) b = σ(c, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, (b−1c)2, [a, b]2, (a−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,33,143,602,2514 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2371 a = σ(a, b) b = σ(c, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (b−1c)2, (a−1b)4, (b−1c−1ac)2, (a−1c)4 SF: 20,21,23,27,213,225,247,290,2176 Gr: 1,7,35,165,758,3460 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Automaton number 2372 a = σ(b, b) b = σ(c, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (a−1b)2, (b−1c)2, [c, ab−1], [cb−1, a], [c−1, b−1] · [a−1, b−1], [a, c−1] · [b, a−1], [b−1, a−1] · [c−1, a−1] SF: 20,21,23,25,27,29,211,213,215 Gr: 1,7,33,127,433,1415 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 2.8 5.6 8.4 11.2 14.0 16.8 19.6 22.4 25.2 28.0 102 Classification of groups generated by automata Automaton number 2375 a = σ(b, c) b = σ(c, a) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (b−1c)2 SF: 20,21,23,25,29,215,226,248,292 Gr: 1,7,35,165,769,3575 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.00 5.40 10.80 16.20 21.60 27.00 32.40 37.80 43.20 48.60 54.00 Automaton number 2391 a = σ(c, b) b = σ(b, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: b2, [a2, b] SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,6,26,103,399,1538 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 9.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2395 a = σ(a, a) b = σ(c, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: a2, c2, (acb)2, [b2, cac] SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,5,17,50,140,377,995,2605 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 Automaton number 2396 a = σ(b, a) b = σ(c, b) c = (c, a) Group: A. Boltenkov group Contracting: no Self-replicating: yes Rels: acb−1ca−2cb−1cac−1bc−2bc−1, acb−1ca−2cb−1a2c−1b−1a2c−1bc−1a−1bca−2bc−1, acb−1a2c−1b−1a−1cb−1cbca−2bc−2bc−1, bcb−1ca−1b−1cb−1a2c−1ac−1ba−2bc−1 SF: 20,21,23,26,212,224,246,290,2176 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 104 Classification of groups generated by automata Automaton number 2398 a = σ(a, b) b = σ(c, b) c = (c, a) Group: F.Dahmani Group Contracting: no Self-replicating: yes Rels: cba, b−1a−1b2a−1b−1a2 SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,7,31,127,483,1823 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.00 4.30 8.60 12.90 17.20 21.50 25.80 30.10 34.40 38.70 43.00 Automaton number 2399 a = σ(b, b) b = σ(c, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: [b−1a, ba−1] SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,929,4599 a b c σ σ 1 0,1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2401 a = σ(a, c) b = σ(c, b) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: (a−1c)2, [a, c]2, (c−2ba)2 SF: 20,21,23,25,29,215,226,248,292 Gr: 1,7,35,165,757,3447 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 3.4 6.8 10.2 13.6 17.0 20.4 23.8 27.2 30.6 34.0 Automaton number 2402 a = σ(b, c) b = σ(c, b) c = (c, a) Group: Contracting: n/a Self-replicating: yes Rels: ac2b−1a−2c2b−1abc−2bc−2, ac2b−1a−2cb−2c−1a4bc−2a−3cb2c−1, acb−2c−1ac2b−1a−2cb2c−1bc−2, acb−2c−1acb−2c−1acb2c−1a−3cb2c−1 SF: 20,21,23,25,27,210,215,225,241 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0 1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 106 Classification of groups generated by automata Automaton number 2423 a = σ(b, a) b = σ(c, c) c = (c, a) Group: Contracting: no Self-replicating: yes Rels: ac−1bca−2c−1bcac−1b−2c, ac−1bca−1c−1bac−1b−1a2c−1b−1ca−1b· ca−1b−1ca−1, bc−1bca−1b−1ac−1bac−1ac−1b−1c2a−1· b−1ca−1, bac−1bac−1b−2c−1bca−1b2ca−1· b−1ca−1b−1ac−1b−1c, bac−1bac−1b−2ac−1bac−1bca−1· b−1ca−1ca−1b−1ca−1 SF: 20,21,23,25,28,214,225,247,290 Gr: 1,7,37,187,937,4687 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 Automaton number 2427 a = σ(c, b) b = σ(c, c) c = (c, a) Group: Contracting: n/a Self-replicating: yes Rels: [b−1a, ba−1], a−1c2a−1b−1a2c−2b SF: 20,21,23,27,213,224,246,289,2175 Gr: 1,7,37,187,929,4583 a b c σ σ 1 0 1 0,1 0 1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Automaton number 2841 a = σ(c, b) b = σ(a, a) c = (c, c) Group: Contracting: no Self-replicating: yes Rels: c, a−1b−1a−2ba−1b−1aba2b−1ab, a−1b−1a−2b−1a−1babab−2abab, a−1ba−1b−2a−1ba−1bab−1a2b−1ab SF: 20,21,23,25,28,213,223,242,279 Gr: 1,5,17,53,161,485, 1457,4359,12991 a b c σ σ 1 0 1 0,1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 1.9 3.8 5.7 7.6 9.5 11.4 13.3 15.2 17.1 19.0 Automaton number 2850 a = σ(c, b) b = σ(b, a) c = (c, c) Group: Contracting: no Self-replicating: yes Rels: c, a−4bab−1a2b−1ab SF: 20,21,23,26,212,223,245,288,2174 Gr: 1,5,17,53,161,485,1445 a b c σ σ 1 0 1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7.0 108 Classification of groups generated by automata Automaton number 2853 a = σ(c, c) b = σ(b, a) c = (c, c) Group: IMG ( ( z−1 z+1 )2 ) Contracting: yes Self-replicating: yes Rels: c, a2, ab−1ab−2ab−1abab2ab SF: 20,21,22,23,25,28,214,225,247 Gr: 1,4,10,22,46,94,190,375,731, 1422,2752,5246,9908 Limit space: a b c σ σ 1 0,1 0 1 0,1 −1.1−0.9−0.7−0.5−0.3−0.10.1 0.3 0.5 0.7 0.9 1.1 0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 9. Proofs This section contains proofs of many of the claims contained in the tables in Section 7 and Section 8 and some additional information. We sometimes encounter one of the following four binary tree auto- morphisms a = σ(1, a), b = σ(b, 1), c = σ(c−1, 1), d = σ(1, d−1). The first one is the binary adding machine, the second is its inverse, and all are conjugate to the adding machine and therefore act level transitively on the binary tree and have infinite order. We freely use the known classification of groups generated by 2-state automata over a 2-letter alphabet. Theorem 7 ([GNS00]). Up to isomorphism, there are six (2, 2)- automaton groups: the trivial group, the cyclic group of order 2 (we denote it by C2), Klein group C2 ×C2 of order 4, the infinite cyclic group Z, the infinite dihedral group D∞ and the Lamplighter group Z ≀ C2. In particular the sixteen 2-state automata for which both states are inactive generate the trivial group, and the sixteen 2-state automata in which both states are active generate C2 (since both states in that case describe the mirror automorphism µ = σ(µ, µ) of order 2. The automata given by either of the wreath recursions a = σ(a, a), b = (a, a), a = σ(b, b), b = (a, a), generate the Klein group C2 × C2. The automata given by the wreath recursions a = σ(a, a), b = (a, b), a = σ(a, a), b = (b, a), a = σ(b, b), b = (a, b), a = σ(b, b), b = (b, a), generate the infinite dihedral group D∞. The automata given by the wreath recursions a = σ(a, a), b = (b, b), a = σ(b, b), b = (b, b), generate the cyclic group C2. 110 Classification of groups generated by automata The automata given by the wreath recursions a = σ(a, b), b = (a, a), a = σ(b, a), b = (a, a), a = σ(a, b), b = (b, b), a = σ(b, a), b = (b, a), generate the infinite cyclic group Z. Moreover, in the first two cases we have b = a−2, in the fourth case b = 1 and a is the adding machine, and in the third case b = 1 and a is the inverse of the adding machine. The automata given by the wreath recursions a = σ(a, b), b = (a, b), a = σ(a, b), b = (b, a), a = σ(b, a), b = (a, b), a = σ(b, a), b = (b, a), generate the Lamplighter group Z ≀ C2 = Z ⋉ (⊕ZC2). The results on the next few pages concern the existence of elements of infinite order and the level transitivity of the action. They are used in some of the proofs that follow. Lemma 1 ([BGK+a]). Let G be a group generated by an automaton A over a 2-letter alphabet. Assume that the set of states S of A splits into two nonempty parts P and Q such that (i) one of the parts consists of the active states (those with nontrivial vertex permutation) and the other consists of the inactive states; (ii) for each state from P , both arrows go to states in the same part (either both to P or both to Q); (iii) for each state from Q, one arrow goes to a state in P and the other to a state in Q. Then any element of the group that can be written as a product of odd number of active generators or their inverses and odd number of inac- tive generators and their inverses, in any order, has infinite order. In particular, the group G is not a torsion group. Proof. Denote by D the set of elements in G that can be represented as a product of odd number of active generators or their inverses and odd number of inactive generators and their inverses, in any order. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, We note that if g ∈ D then both sections of g2 are in D. Indeed, for such an element, g = σ(g0, g1) and g2 = (g1g0, g0g1). Both sections of g2 are products (in some order) of the first level sections of the generators (and/or their inverses) used to express g as an element in D. By assump- tion, among these generators, there are odd number of active and odd number of inactive ones. The generators from P , by condition (ii), pro- duce even number of active and even number of inactive sections on level 1, while the generators from Q, by condition (iii), produce odd number of active sections and odd number of inactive sections. Thus both sections of g are in D. By way of contradiction, assume that h is an element of D of finite order 2n, for some n ≥ 0. If n > 0 the sections of h2 are elements in D of order 2n−1. Thus, continuing in this fashion, we reach an element in D that is trivial. This is contradiction since all elements in D act nontrivially on level 1. There is a simple criterion that determines whether a given element of a self-similar group generated by a finite automaton over the 2-letter alphabet X = {0, 1} acts level transitively on the tree. The criterion is based on the image of the given element in the abelianization of Aut(X∗), which is isomorphic to the infinite Cartesian product ∏∞ i=0C2. The canonical isomorphism sends g ∈ G to (ai mod 2)∞i=0, where ai is the number of active sections of g at level i. We also make use of the ring structure on ∏∞ i=0C2 obtained by identifying (bi) ∞ i=0 with ∑∞ i=0 bit i in the ring of formal power series C2[[t]]. It is known that a binary tree auto- morphism g acts level transitively on X∗ if and only if ḡ = (1, 1, 1, . . .), where ḡ be the image of g in the abelianization ∏∞ i=0C2 of Aut(X∗). Lemma 2 (Element transitivity, [BGK+a]). Let G be a group generated by an automaton A over a 2-letter alphabet. There exists an algorithm that decides if g acts level transitively on X∗. Proof. Let g = σi(g0, g1), where i ∈ {0, 1}. Then g = i+ t · (g0 + g1). Similar equations hold for all sections of g. SinceG is generated by a finite automaton, g has only finitely many different sections, say k. Therefore we obtain a linear system of k equations over the k variables {gv, v ∈ X∗}. The solution of this system expresses ḡ as a rational function P (t)/Q(t), where P an Q are polynomials of degree not higher than k. The element g acts level transitively if and only if ḡ = 1 1−t . 112 Classification of groups generated by automata We often need to show that a given group of tree automorphisms is level transitive. Here is a very convenient necessary and sufficient condition for this in the case of a binary tree. Lemma 3 (Group transitivity, [BGK+a]). A self-similar group of binary tree automorphisms is level transitive if and only if it is infinite. Proof. Let G be a self-similar group acting on a binary tree. If G acts level transitively then G must be infinite (since the size of the levels is not bounded). Assume now that the group G is infinite. We first prove that all level stabilizers StabG(n) are different. Note that, since all level stabilizers have finite index in G and G is infinite, all level stabilizers are infinite. In particular, each contains a nontrivial element. Let n > 0 and g ∈ StabG(n − 1) be an arbitrary nontrivial element. Let v = x1 . . . xk be a word of shortest length such that g(v) 6= v. Since g ∈ StabG(n − 1), we must have k ≥ n. The section h = gx1x2...xk−n is an element of G by the self-similarity of G. The minimality of the word v implies that g ∈ StabG(k − 1), and therefore h ∈ StabG(n − 1). On the other hand h acts nontrivially on xk−n+1 . . . xk and we conclude that h ∈ StabG(n− 1) \ StabG(n). Thus all level stabilizers are different. We now prove level transitivity by induction on the level. The existence of elements in StabG(0) \ StabG(1) shows that G acts transitively on level 1. Assume thatG acts transitively on level n. Select an arbitrary element h ∈ StabG(n) \ StabG(n + 1) and let w =∈ Xn be a word of length n such that h(w1) = w0. Let u be an arbitrary word of length n and let x be a letter in X = {0, 1}. We will prove that ux is mapped to w0 by some element of G, proving the transitivity of the action at level n + 1. By the inductive assumption there exists f ∈ G such that f(u) = w. If f(ux) = w0 we are done. Otherwise, hf(ux) = h(w1) = w0 and we are done again. Consider the infinitely iterated permutational wreath product ≀i≥1Cd, consisting of the automorphisms of the d-ary tree for which the activity at every vertex is a power of some fixed cycle of length d. The last proof works, mutatis mutandis, for the self-similar subgroups of ≀i≥1Cd and may be easily adapted in other situations. The following lemma is used often when we want to prove that some automaton group is not free. Lemma 4. If a self-similar group contains two nontrivial elements of the form (1, u), (v, 1), then the group is not free. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Proof. Suppose a = (1, u), b = (v, 1) are two nontrivial elements of a self-similar group G and G is free. Obviously [a, b] = 1, hence a and b are powers of some element x ∈ G: a = xm, b = xn. Then an = bm, so an = (1, un) = bm = (vm, 1). This implies that un = vm = 1, which is a contradiction, since u and v are nontrivial elements of a free group. In most case when the corresponding group is finite we do not offer a full proof. In all such cases the proof can be easily done by direct calculations. As an example, a detailed proof is given in the case of the automaton [748]. We now proceeds to individual analysis of the properties of the au- tomaton groups in our classification. 1. Trivial group. 730. Klein Group C2 × C2. Wreath recursion: a = σ(a, a), b = (a, a), c = (a, a). The claim follows from the relations b = c, a2 = b2 = abab = 1. 731 ∼= Z. Wreath recursion: a = σ(b, a), b = (a, a), c = (a, a). We have c = b and b = a−2. The states a and b form a 2-state automaton generating Z (see Theorem 7). 734 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(b, b), b = (a, a), c = (a, a). The claim follows from the relations b = c, a2 = b2 = abab = 1. 739 ∼= C2 ⋉ (C2 ≀Z). Wreath recursion: a = σ(a, a), b = (b, a), c = (a, a). All generators have order 2. The elements u = acba = (1, ba) and v = bc = (ba, 1) generate Z 2. This is clear since ba = σ(1, ba) is the adding machine and therefore has infinite order. Further, we have ac = σ and 〈u, v〉 is normal in H = 〈u, v, σ〉, since uσ = v and vσ = u. Thus H ∼= C2 ⋉ (Z × Z) = C2 ≀ Z. We have G739 = 〈H, a〉 and H is normal in G739, since it has index 2. Moreover, ua = v−1, va = u−1 and σa = σ. Thus G739 = C2 ⋉ (C2 ≀ Z), where the action of C2 on H is specified above. 740. Wreath recursion: a = σ(b, a), b = (b, a), c = (a, a). The states a, b form a 2-state automaton generating the Lamplighter group (see Theorem 7). Thus G740 has exponential growth and is neither torsion nor contracting. Since c = (a, a) we obtain that G740 can be embedded into the wreath product C2 ≀ (Z ≀ C2). Thus G740 is solvable. 741. Wreath recursion: a = σ(c, a), b = (b, a), c = (a, a). The states a and c form a 2-state automaton generating the infinite cyclic group Z in which c = a−2 (see Theorem 7). Since b = (b, a), we see that b has infinite order and that G741 is not contracting). 114 Classification of groups generated by automata We have c = a−2 and b−1a−3b−1ababa = 1. Since a and b do not commute the group is not free. 743 ∼= G739 ∼= C2 ⋉ (C2 ≀ Z). Wreath recursion: a = σ(b, b), b = (b, a), c = (a, a). All generators have order 2. The elements u = acba = (1, ba) and v = bc = (ba, 1) generate Z 2 because ba = σ(ab, 1) is conjugate to the adding machine and has infinite order. Further, we have babc = σ and 〈u, v〉 is normal in H = 〈u, v, σ〉 because uσ = v and vσ = u. In other words, H ∼= C2 ⋉ (Z × Z) = C2 ≀ Z. Furthermore, G743 = 〈H, a〉 and H is normal in G743 because ua = v−1, va = u−1 and σa = σ. Thus G743 = C2⋉(C2 ≀Z), where the action of C2 on H is specified above and coincides with the one in G739. Therefore G743 ∼= G739. 744. Wreath recursion: a = σ(c, b), b = (b, a), c = (a, a). Since (a−1c)2 = (c−1ab−1a, b−1ac−1a) and c−1ab−1a = ((c−1ab−1a)−1, a−1c), the element (a−1c)2 fixes the vertex 01 and its section at this vertex is equal to a−1c. Hence, a−1c has infinite order. The element c−1ab−1a also has infinite order, fixes the vertex 00 and its section at this vertex is equal to c−1ab−1a. Therefore G744 is not contracting. We have b−1c−1ba−1ca = (1, a−1c−1ac), ab−1c−1ba−1c = (ca−1c−1a, 1), hence by Lemma 4 the group is not free. 747 ∼= G739 ∼= C2 ⋉ (C2 ≀ Z). Wreath recursion: a = σ(c, c), b = (b, a), c = (a, a). All generators have order 2 and a commutes with c. Conjugating this group by the automorphism γ = (γ, cγ) yields an isomorphic group generated by automaton a′ = σ, b′ = (b′, a′) and c′ = (a′, a′). On the other hand we obtain the same automaton after conjugating G739 by µ = (µ, aµ) (here a denotes the generator of G739). 748 ∼= D4 × C2. Wreath recursion: a = σ(a, a), b = (c, a), c = (a, a). Since a is nontrivial and b and c have a as a section, none of the generators is trivial. All generators have order 2. Indeed, we have a2 = (a2, a2), b2 = (c2, a2), c2 = (a2, a2), showing that a2, b2 and c2 generate a self-similar group in which no element is active. Therefore a2 = b2 = c2 = 1. Since ac = σ we have that (ac)2 = 1. Therefore a and c commute. Since (bc)2 = ((ca)2, 1) = 1, we see that b and c also commute. Further, the relations (ab)2 = (ac, 1) = (σ, 1) 6= 1 and (ab)4 = 1 show that a and b generate the dihedral group D4. It remains to be shown that c 6∈ 〈a, b〉. Clearly c could only be equal to one of the four elements 1, b, aba, and abab in D4 that stabilize level 1. However, c is nontrivial, differs from b at 0 (the section b\|0 = c is not active, while c\|0 = a is active), differs from aba at 1 (the section (aba)\|1 = aca is not active, while c\|1 = a is I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, active), and differs from abab at 1 (the section of abab at 1 is trivial). This completes the proof. 749. Wreath recursion: a = σ(b, a), b = (c, a), c = (a, a). The element (a−1c)4 stabilizes the vertex 000 and its section at this vertex is equal to a−1c. Hence, a−1c has infinite order. We have ac−1 = σ(ba−1, 1), ba−1 = σ(1, cb−1), cb−1 = (ac−1, 1), Thus the subgroup generated by these elements is isomorphic to IMG(1− 1 z2 ) (see [BN06]). We have c−1b = (a−1c, 1), ac−1ba−1 = (1, ca−1). Thus, by Lemma 4 the group is not free. 748 ∼= G848 ∼= C2 ≀Z. Wreath recursion:a = σ(c, a), b = (c, a), c = (a, a). It is proven below that G848 ∼= G2190 and for G2190 we have a = σ(c, a), b = σ(a, a), c = (a, a). Therefore G2190 = 〈a, b, c〉 = 〈a, c, c−1b = σ〉 = 〈a = (c, a)σ, c = (a, a), aσ = (c, a)〉 = G750. 752. Wreath recursion: a = σ(b, b), b = (c, a), c = (a, a). The group G752 is a contracting group with nucleus consisting of 41 elements. It is a virtually abelian group, containing Z 3 as a subgroup of index 4. All generators have order 2. Let x = ca, y = babc, and K = 〈x, y〉. Since xy = ((cbab)ca, abcb) = ((y−1)x, abcb) and yx = (cbab, abcb) = (y−1, abcb) the elements x and y commute. Conjugating by γ = (γ, bcγ) yields the self-similar copy K ′ of K generated by x′ = σ((y′)−1, (x′)−1) and y′ = σ((y′)−1x′, 1), where x′ = xγ and y′ = yγ . Since (x′)2 = ((x′)−1(y′)−1, (y′)−1(x′)−1) and (y′)2 = ((y′)−1x′, (y′)−1x′), the virtual endomorphism of K ′ is given by A = ( −1 2 1 2 −1 2 −1 2 ) . The eigenvalues λ = −1 2 ± 1 2 i of this matrix are not algebraic integers, and therefore, by the results in [NS04], the group K ′ ∼= K is free abelian of rank 2. Let H = 〈ba, cb〉. The index of StabH(1) in G is 4, since the index of StabH(1) in H is 2 and the index of H in G is 2 (the generators have order 2). We have StabH(1) = 〈cb, cbba, (ba)2〉. If we conjugate the generators of StabH)(1) by g = (1, b), we obtain g1 = (cb)g = (x−1,1), g2 = ( (cb)ba )g = (1, x), g3 = ( (ba)2 )g = (y−1,y). 116 Classification of groups generated by automata Therefore, g1, g2, and g3 commute. If gn1 1 gn2 2 gn3 3 = 1, then we must have x−n1y−n3 = xn2yn3 = 1. Since K is free abelian, this implies n1 = n2 = n3 = 0. Thus, StabH(1) is a free abelian group of rank 3. 753. Wreath recursion: a = σ(c, b), b = (c, a), c = (a, a). Since ab−1 = σ(1, ba−1), this element is conjugate to the adding ma- chine. For a word w in w ∈ {a±1, b±1, c±1}∗, let \|w\|a, \|w\|b and \|w\|c denote the sum of the exponents of a, b and c in w. Let w represents the element g ∈ G. If \|w\|a and \|w\|b are odd, then g acts transitively on the first level, and g2\|0 is represented by a word w0, which is the product (in some order) of all first level sections of all generators appearing in w. Hence, \|w0\|a = \|w\|b + 2\|w\|c and \|w0\|b = \|w\|a are odd again. Therefore, similarly to Lemma 1, any such element has infinite order. In particular c2ba has infinite order. Since a4 = (caca, a4, acac, a4) and caca = (baca, c2ba, bac2, caba), the element a4 has infinite order (and so does a). Since a4 fixes the vertex 01 and its section at that vertex is equal to a4, the group G753 is not contracting. We have cb−1 = (ac−1, 1), acb−1a−1 = (1, bac−1b−1), hence by Lemma 4 the group is not free. 756 ∼= G748 ∼= D4 × C2. Wreath recursion: a = σ(c, c), b = (c, a), c = (a, a). All generators have order 2. The generator c commutes with both a and b. Since (ab)2 = (ca, ca) the order of ca is 4 and the group is isomorphic to D4 × C2 766 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(a, a), b = (b, b), c = (a, a). The state b is trivial. The states a and c form a 2-state automaton generating C2 × C2 (see Theorem 7). 767 ∼= G731 ∼= Z. Wreath recursion: a = σ(1, a), b = (b, b), c = (a, a) = a2. The state b is trivial. The automorphism a is the binary adding machine. 768 ∼= G731 ∼= Z. Wreath recursion: a = σ(c, a), b = (b, b), c = (a, a). The states a and c form a 2-state automaton generating Z (see The- orem 7) in which c = a−2. 770 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(b, b), b = (b, b), c = (a, a). The state b is trivial. The states a and c form a 2-state automaton generating C2 × C2 (see Theorem 7). 771 ∼= Z 2. Wreath recursion: a = σ(c, b), b = (b, b), c = (a, a). The group G771 is finitely generated, abelian, and self-replicating. Therefore, it is free [NS04]. Since b = 1 the rank is 1 or 2. We prove I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, that the rank is 2, by showing that cn 6= am, unless n = m = 0. By way of contradiction, let cn = am for some integer n and m and choose such integers with minimal \|n\| + \|m\|. Since cn stabilizes level 1, m must be even and we have (an, an) = cn = am = (cm/2, cm/2), implying an = cm/2. By the minimality assumption, m must be 0, which then implies that n must be 0 as well. 774 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(c, c), b = (b, b), c = (a, a). The state b is trivial. The states a and c form a 2-state automaton generating C2 × C2 (see Theorem 7). 775 ∼= C2 ⋉ IMG ( ( z−1 z+1 )2 ) . Wreath recursion: a = σ(a, a), b = (c, b), c = (a, a). All generators have order 2. Further, ac = ca = σ(1, 1) and ba = σ(ba, ca). Hence, for the subgroup H = 〈ba, ca〉 ∼= G2853 ∼= IMG ( ( z−1 z+1 )2 ) . Since the generators have order 2, H is normal subgroup of index 2 in G775. Moreover (ba)a = (ba)−1 and (ca)a = ca. Therefore G ∼= C2 ⋉H, where C2 is generated by a and the action of a on H is given above. Conjugating the generators by g = σ(g, g) we obtain the wreath re- cursion a′ = σ(a′, a′), b′ = (b′, c′), c′ = (a′, a′), where a′ = ag, b′ = bg and c′ = cg. This is the wreath recursion defining G793. Denote G793 by G and its generators by a, b, and c (we continue working only with G793). Thus a = σ(a, a), b = (b, c), c = (a, a). The generators have order 2. Moreover ac = ca and 〈a, c〉 = C2 × C2 is the Klein group. Denote A = 〈a, c〉. The element x = ba has infinite order, since x2 fixes 00, and has itself as a section at 00. Note that x = ba = (b, c)σ(a, a) = σ(ca, ba) = σ(σ, x). and, therefore, x2 = (xσ, σx) = (x, σ, σ, x). Proposition 1. The subgroup H = 〈x, y〉 of G, where x = ba and y = cabc is torsion free. Proof. The first level decompositions of x±1 and y±1 and the second level 118 Classification of groups generated by automata decompositions of x and y are given by x = σ(σ, x) y = cabc = σaabaσ = σbaσ = xσ = σ(x, σ) x−1 = σ(x−1, σ) y−1 = σ(σ, x−1) x = σ(σ(1, 1), σ(σ, x)) = µ(1, 1, σ, x) y = xσ = µ(σ, x, 1, 1), where µ = σ(σ, σ) permutes the first two levels of the tree as 00 ↔ 11, 10 ↔ 01. We encode this as the permutation µ = (03)(12). For a word w over {x±1, σ}, denote by #x(w) and #σ(w) the total number of appearances of x and x−1 and the number of appearances of σ in w, respectively. Note that x and x−1 act as the permutation (03)(12) on the sec- ond level, and σ acts as the permutation (02)(13). These permutations have order 2, commute, and their product is (01)(23), which is not triv- ial. Thus, a tree automorphisms represented by a word w over {x±1, σ} cannot be trivial unless both #x(w) and #σ(w) are even. Let g be an element of H that can be written as g = z1z2 . . . zn, for some zi ∈ {x±1, y±1}, i = 1, . . . , n. If n is odd, the element g cannot have order 2. By way of contradiction assume otherwise. For z in {x±1, y±1} denote z′ = σz. Thus, for instance x′ = (σ, x) and y′ = (x, σ). Note that g2 = (z1z2 . . . zn)2 = (z′1) σz′2(z ′ 3) σz′4 . . . (z ′ n)σz′1(z ′ 2) σ . . . z′n = (w0, w1), where the words wi over {x±1, σ} are such that #x(wi) = #σ(wi) = n, (8) for i = 1, 2. The last claim holds because exactly one of z′i and (z′i) σ con- tributes x±1 to w0 and σ to w1, respectively, while the other contributes the same letters to w1 and w0, respectively. Since n is odd, (8) shows that neither w0 nor w1 can be 1 and therefore g2 cannot be 1. Assume that H contains an element of finite order. In particular, this implies that H must contain an element of order 2. Let g = z1z2 . . . zn be such an element of the shortest possible length, where zi ∈ {x±1, y±1}, i = 1, . . . , n. Note that n must be even. Therefore, g = z1z2 . . . zn = (z′1) σz′2 . . . (z ′ n−1) σz′n = (w0, w1), I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, where w0 and w1 are words over {x±1, σ}. Moreover, as elements in H, the orders of w0 and w1 divide 2 and the order of at least one of them is 2. We claim that #x(w0) ≡ #σ(w0) ≡ #x(w1) ≡ #σ(w0) mod 2. (9) The congruence #x(wi) ≡ #σ(wi) mod 2 holds because #x(wi) + #σ(wi) = n is even. For the other congruences, observe that whenever z′i or (z′i) σ contributes x±1 or σ to w0, respectively, it contributes σ or x±1 to w1, respectively. Therefore #x(w0) = #σ(w1) and #σ(w0) = #x(w1). If the numbers in (9) are even, then w0 and w1 represent elements in H and can be rewritten as words over {x±1, y±1} of lengths at most #x(w0) = n − #σ(w0) and #x(w1) = n − #σ(w1), respectively. If both of these lengths are shorter than n then none of them can represent an element of order 2 in H. Otherwise, one of the words wi is a power of x and the other is trivial. Sice x has infinite order this shows that g cannot have order 2. If the numbers in (9) are odd, then, for i = 1, 2, wi can be rewritten as σui, where ui are words of odd length over {x±1, y±1}. Let w0 = σt1 . . . tm, where m is odd, and tj are letters in {x±1, y±1}, j = 1, . . . ,m. We have w0 = t′1(t ′ 2) σ . . . (t′m−1) σt′m = (w00, w01), where w00 and w01 are words of odd length m over {x±1, σ}. Moreover, exactly one of the words w00 and w01 has even number of σ’s and this word can be rewritten as a word over {x±1, y±1} of odd length. However, an element in H represented by such a word cannot have order dividing 2. This completes the proof. Since xa = abaa = ab = x−1, ya = acabca = cbac = y−1, xb = bbab = ab = x−1, yb = bcabcb = bacbacab = xy−1x−1, xc = cbac = y−1, yc = ccabcc = ab = x−1, we see that H is the normal closure of x in G. Further, G = {x, y, a, c} and G = AH. It follows from Proposition 1 that A ∩H = 1 (since A is finite) and therefore G = A⋉H. Proposition 2. The group G is a regular, weakly branch group, branching over H ′′. Proof. The group G is infinite self-similar group acting on a binary three. Therefore it is level transitive by Lemma 3. 120 Classification of groups generated by automata Since x2 = (x, σ, σ, x) y−1x2y = (y, x−1σx, σ, x) we have that H ′′ × 〈σ, x−1σx〉′′ × 〈σ〉′′ × 〈x〉′′ � H ′′. On the other hand, 〈σ, x−1σx〉 is metabelian (in fact dihedral, since the generators have order 2) and 〈σ〉 and 〈x〉 are abelian (cyclic). There- fore H ′′ × 1 × 1 × 1 � H ′′. The group H ′′ is normal in G, since it is characteristic in the normal subgroup H. Finally, H ′′ is not trivial. For instance it is easy to show that [[x, y], [x, y−1]] 6= 1 (see [BGK+b]). 776. Wreath recursion: a = σ(b, a), b = (c, b), c = (a, a). The element (b−1a)4 stabilizes the vertex 00 and its section at this vertex is equal to (b−1a)−1. Hence, b−1a has infinite order. Furthermore, by Lemma 1 ab has infinite order, which yields that a,c and b also have infinite order, because a2 = (ab, ba). Since bn = (cn, bn) we obtain that bn belong to the nucleus for all n ≥ 1. Thus G776 is not contracting. We have a−1ba−1c = (1, b−1c), ba−1ca−1 = (cb−1, 1), hence by Lemma 4 the group is not free. 777. Wreath recursion: a = σ(c, a), b = (c, b), c = (a, a). The states a, c form the 2-state automaton generating Z (see Theo- rem 7). So the group is not torsion and G777 = 〈a, b〉. Since c has infinite order, so has b. Therefore the relation bn = (cn, bn) implies that bn belong to the nucleus for all n ≥ 1. Thus G777 is not contracting. Also we have ab−1 = σ(1, ab−1) is the adding machine. Since a−3 = σ(1, a3) elements ab−1 and a−3 generate the Brunner-Sidki-Vierra group (see [BSV99]). 779. Wreath recursion: a = σ(b, b), b = (c, b), c = (a, a). The element (ab−1)2 stabilizes the vertex 01 and its section at this vertex is equal to (ab−1)−1. Hence, ab−1 has infinite order. 780. Wreath recursion: a = σ(c, b), b = (c, b), c = (a, a). The element (c−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to c−1a. Hence, c−1a has infinite order. Since [c, a] ∣ ∣ 100 = (c−1a)a and 100 is fixed under the action of [c, a] we obtain that [c, a] also has infinite order. Finally, [c, a] stabilizes the vertex 00 and its section at this vertex is [c, a]. Therefore G780 is not contracting. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 783 ∼= G775 ∼= C2 ⋉ IMG ( ( z−1 z+1 )2 ) . Wreath recursion: a = σ(c, c), b = (c, b), c = (a, a). All generators have order 2 and ac = ca. If we conjugate the genera- tors of this group by the automorphism γ = (cγ, γ) we obtain the wreath recursion a′ = σ(1, 1), b′ = (c′, b′), c′ = (a′, a′), where a′ = aγ , b′ = bγ , and c′ = cγ . The same wreath recursion is obtained after conjugating G775 by µ = (aµ, µ) (where a denotes the generator of G775). Since bca = σ(bca, a), G783 = 〈acb, a, c〉 ∼= G2205. 802 ∼= C2×C2×C2. Wreath recursion: a = σ(a, a), b = (c, c), c = (a, a). Direct calculation. 803 ∼= G771 ∼= Z 2. Wreath recursion: a = σ(b, a), b = (c, c), c = (a, a). The group G771 is finitely generated, abelian, and self-replicating. Therefore, it is free abelian [NS04]. Let φ : StabG803 (1) → G803 be the 1 2 -endomorphism associated to the vertex 0, given by φ(g) = h, provided g = (h, ∗). The matrix of the linear map C 3 → C 3 induced by φ with to the basis corresponding to the triple {a, b, c} is given by A = ( 1 2 0 1 1 2 0 0 0 1 0 ) . The eigenvalues are λ1 = 1, λ2 = −1 4− 1 4 i √ 7 and λ3 = −1 4 + 1 4 i √ 7. Let vi, i = 1, 2, 3, be eigenvectors corresponding to the eigenvalues λi, i = 1, 2, 3. Note that v1 may be selected to be equal to v1 = (2, 1, 1). This shows that a2bc = 1 in G803 and the rank of G803 = 〈a, c〉 is at most 2. We will prove that a2mcn 6= 1 (except when m = n = 0) by proving that iterations of the action of A eventually push the vector v = (2m, 0, n) out of the set D = {(2i, j, k), i, j, k ∈ Z} corresponding to the first level stabilizer. Let v = a1v1 + a2v2 + a3v3. The vector v is not a scalar multiple of v1. Therefore either a2 6= 0 or a3 6= 0. Since \|λ2\| = \|λ3\| < 1, we have At(v) = a1v1 + λt 2a2v2 + λt 3a3v3 → a1v1, as t → ∞. Note that, since a2 6= 0 or a3 6= 0, At(v)is never equal to a1v1. Choose a neighborhood U of a1v1 that does not contain vectors from D, except possibly the vector a1v1. For t large enough t, the vector At(v) is in U and is therefore outside of D. Thus the rank of G803 is 2. 804 ∼= G731 ∼= Z. Wreath recursion: a = σ(c, a), b = (c, c), c = (a, a). Indeed, the states a and c form a 2-state automaton generating the cyclic group Z (see Theorem 7). Since b = a4 we are done. 122 Classification of groups generated by automata 806 ∼= G802 ∼= C2 × C2 × C2. Wreath recursion: a = σ(b, b), b = (c, c), c = (a, a). Direct calculation. 807 ∼= G771 ∼= Z 2. Wreath recursion: a = σ(c, b), b = (c, c), c = (a, a). The same arguments as in the case of G771 show that G807 is free abelian. It has a relation c2ba2 = 1 and, hence, it has either rank 1 or rank 2. Analogically to G803 we consider a 1 2 -endomorphism φ : StabG807 (1) → G807, and a linear map A : C 3 → C 3 induced by φ. It has the following matrix representation with respect to the basis corresponding to the triple {a, b, c}: A = ( 0 0 1 1 2 0 0 1 2 1 0 ) . Its characteristic polynomial χA(λ) = −λ3 + 1 2λ+ 1 2 has three distinct complex roots λ1 = 1, λ2 = −1 2 − 1 2 i and λ3 = −1 2 + 1 2 i. Analogically for v = (2m, 0, n) we get that At(v) will be pushed out from the domain corresponding to StabG807 (1). Thus cna2m 6= 1 in G807 and G807 ∼= Z 2. 810 ∼= G802 ∼= C2 × C2 × C2. Wreath recursion: a = σ(c, c), b = (c, c), c = (a, a). Direct calculation. 820 ∼= D∞. Wreath recursion: a = σ(a, a), b = (b, a), c = (b, a). The states a and b form a 2-state automaton generating D∞ (see Theorem 7) and c = b. 821. Lamplighter group Z ≀C2. Wreath recursion: a = σ(b, a), b = (b, a), c = (b, a). The states a and b form a 2-state automaton generating the Lamp- lighter group (see Theorem 7) and c = b. 824 ∼= G820 ∼= D∞. Wreath recursion: a = σ(a, a), b = (b, a), c = (b, a). The states a and b form a 2-state automaton generating D∞ (see Theorem 7) and c = b. 838 ∼= C2 ⋉ 〈s, t ∣ ∣ s2 = t2〉. Wreath recursion: a = σ(a, a), b = σ(a, b), c = (b, a). All generators have order 2. Consider the subgroup H = 〈ba = σ(ba, 1), ca = σ(1, ab)〉 ∼= G2860 = 〈s, t ∣ ∣ s2 = t2〉. This subgroup is nor- mal in G838 because the generators have order 2. Since G838 = 〈H, a〉, it has a structure of a semidirect product 〈a〉 ⋉ H = C2 ⋉ 〈s, t ∣ ∣ s2 = t2〉 with the action of a on H as (ba)b = (ba)−1 and (ca)b = (ca)−1. 839 ∼= G821. Lamplighter group Z ≀ C2. Wreath recursion: a = σ(b, a), b = (a, b), c = (b, a). The states a and b form a 2-state automaton generating the Lamp- lighter group (see Theorem 7). Since b−1a = σ = ac−1, we see that I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, c = a−1ba and G = 〈a, b〉. 840. Wreath recursion: a = σ(c, a), b = (a, b), c = (b, a). The element (b−1a)2 stabilizes the vertex 01 and its section at this vertex is equal to b−1a. Hence, b−1a has infinite order. The element (c−1b)2 stabilizes the vertex 00 and its section at this vertex is equal to (c−1b)−1. Hence, c−1b has infinite order. Since (b−1a−1b−1cba)2 ∣ ∣ 00000000 = c−1b and the vertex 00000000 is fixed under the action of (b−1a−1b−1cba)2 we obtain that b−1a−1b−1cba also has infi- nite order. Finally, b−1a−1b−1cba stabilizes the vertex 0001 and has itself as a section at this vertex. Therefore G840 is not contracting. We have b−1a−1ca = (1, b−1c−1bc), ab−1a−1c = (cb−1c−1b, 1), hence by Lemma 4 the group is not free. 842 ∼= G838 ∼= C2 ⋉ 〈s, t ∣ ∣ s2 = t2〉. Wreath recursion: a = σ(b, b), b = σ(a, b), c = (b, a). All generators have order 2. Consider the subgroup H = 〈u = ba = σ(1, ba) = σ(1, u−1), v = ca = σ(ab, 1) = σ(u−1, 1)〉. Let us prove that H ∼= W = 〈s, t ∣ ∣ s2 = t2〉. Indeed, the relation u2 = v2 is satisfied, so H is a homomorphic image of W with respect to the homomorphism induced by s 7→ u and t 7→ v. Each element of W can be written in its normal form tr(st)lsn, n ∈ Z, l ≥ 0, r ∈ {0, 1}. It suffices to prove that images of these words (except for the identity word, of course) represent nonidentity elements in H. We have u2n = (u−n, u−n), u2n+1 = σ(a−n, a−n−1) for any integer n; (uv)l = (u2l, 1) for any integer l. Thus (uv)lu2n = (u−2l−n, u−n) 6= 1 in G if n 6= 0 or l 6= 0 since u has infinite order, as it is conjugate to the adding machine. Furthermore, v(uv)lu2n = σ(u−2l−n−1, u−n) 6= 1, (uv)lu2n+1 = σ(u−n, u−2l−n−1) 6= 1 since they act nontrivially on the first level of the tree. Finally, v(uv)lu2n+1 = (u−2l−n−2, u−n) = 1 if and only if n = 0 and l = −1, which is not the case, because l must be nonnegative. Thus H ∼= W . The subgroup H is normal in G842 because generators are of order 2. Since G842 = 〈H, a〉, it has a structure of a semidirect product 〈a〉⋉H = C2 ⋉ 〈s, t ∣ ∣ s2 = t2〉 with the action of a on H as (ba)b = (ba)−1 and (ca)b = (ca)−1. Therefore it has the same structure as G838. 124 Classification of groups generated by automata 843. Wreath recursion: a = σ(c, b), b = (a, b), c = (b, a). The element c−1a = σ(a−1c, 1) is a conjugate of the adding machine. Therefore, it acts transitively on the level of the tree and has infinite order. Since (c−1ab−1a)2 fixes the vertex 000 and its section at this vertex is equal to c−1a, we obtain that c−1ab−1a has infinite order. Since the element c−1ab−1a fixes the vertex 10 and has itself as a section at this vertex, G843 is not contracting. We have c−1a−1ba = (1, a−1c−1ac), ac−1a−1b = (ca−1c−1a, 1), hence by Lemma 4 the group is not free. 846 ∼= C2 ∗ C2 ∗ C2. Wreath recursion: a = σ(c, c), b = (a, b), c = (b, a). The automaton [846] was studied during the Advanced Course on Automata Groups in Bellaterra, Spain, in the summer of 2004 and is since called the Bellaterra automaton. We present here a proof that G846 = C2 ∗C2 ∗C2, based on the concept of dual automata. A different proof, still based on dual automata, is given in [Nek05]. Let A = (Q,X, π, τ) be a finite automaton. Its dual automaton, by definition, is A′ = (X,Q, π′, τ ′), where π′(x, q) = τ(q, x), and τ ′(x, q) = π(q, x). Thus the dual automaton is obtained by exchanging the roles of the states and the alphabet (and the roles of the transition and output function) in a given automaton. The notion od dual automata is not new, but there is a recent renewed interest based on the new results and applications in [MNS00, GM05, BŠ06, VV05]. If in addition to A, both A′ and (A−1)′ are invertible, the automa- ton A is called fully invertible (or bi-reversible). Examples of such au- tomata are the automaton 2240 generating a free group with three gen- erators [VV05], Bellaterra automaton [846], and various automata con- structed in [GM05], generating free groups of various ranks. We now consider the automaton [846] and its dual more closely. Since the generators a, b, and c have order 2, in order to prove that G846 ∼= C2 ∗ C2 ∗ C2 we need to show that no word in w ∈ Rn, n ≥ 1, is trivial in G846, where Rn is the set of reduced words over {a, b, c} of length n (here a word is reduced if it does not contain aa, bb, or cc). For every n > 0, the set of words in Rn that are nontrivial in G846 is nonempty, since the word rn = acbcbcb · · · of length n acts nontrivially on level 1. If we prove that the dual automaton acts transitively on the sets Rn, n ≥ 1, this would mean that rn is a section of every element of G846 that can be represented as a reduced word of length n. Therefore, every word in Rn would represent a nontrivial element in G846 and our proof would be complete. The automaton dual to 846 is the invertible automaton defined by I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, the wreath recursion A = (acb)(B,A,A), B = (ac)(A,B,B), (10) where the three coordinates in the recursion represent the sections at a, b, and c, respectively. Denote D = 〈A,B〉. The set R = ⋃ n≥0Rn of all reduced words over {a, b, c} is a subtree of the ternary tree {a, b, c}∗ and this subtree R is invariant under the action of D (this is because the set {aa, bb, cc} is invariant under the action of D). The structure of R is as follows. The root of R has three children a, b and c, each of which is a root of a binary tree. We want to understand the actio of D on the subtree R. It is given by A = (acb)(Ba, Ab, Ac) B = (ac)(Aa, Bb, Bc) (11) where Aa, Ab, Ac, Ba, Bb, Bc are automorphisms of the binary trees hang- ing down from the vertices a, b and c. After identification of these three trees with the binary tree {0, 1}∗, the action of Aa, Ab, . . . , Bc is defined by Aa = (Ab, Ac), Ab = σ(Ba, Ac), Ac = σ(Ba, Ab), Ba = σ(Bb, Bc), Bb = σ(Aa, Bc), Bc = σ(Aa, Bb). (12) Using Lemma 2 one can verify that Bb acts level transitively on the binary tree. This is sufficient to show that D acts transitively on R, since it acts transitively on the first level, B stabilizes the vertex b, and its section at b is Bb. The fact that G846 is not contracting follows now from the result of Nekrashevych [Nek07a], that a contracting group can not have free subgroups. Alternatively, it is sufficient to observe that aba has infinite order, stabilizes the vertex 01 and has itself as a section at this vertex. 847 ∼= D4. Wreath recursion: a = σ(a, a), b = (b, b), c = (b, a). The state b is trivial. The states a and c form a 2-state automaton generating D4 (see Theorem 7). 848 ∼= C2 ≀ Z. Wreath recursion: a = σ(b, a), b = (b, b), c = (b, a). The state b is trivial and a is the adding machine. Every element g ∈ G848 has the form g = σi(an, am). On the other hand, c = (1, a), cac−1 = (a, 1), so StabG(1) = {(an, am)} ∼= Z 2. Since ac−1 = σ we see that G ∼= C2 ≀ Z. 849. Wreath recursion: a = σ(c, a), b=(b,b), c = (b, a). 126 Classification of groups generated by automata The state b is trivial. The element a2c = (ac, ca2) is nontrivial be- cause its section at 0 is ac, and ac acts nontrivially on level 1. The automorphism (a2c)2 fixes the vertex 00 and its section at this vertex is equal to a2c. Therefore a2c has infinite order. Further, the section of a2c at 100 coincides with a2c, implying that G849 is not contracting. The group G849 is regular weakly branch group over its commutator G′ 849. This is clear since the group is self-replicating and [a−1, c] · [c, a] = ([a, c], 1). Conjugation of the generators of G849 by µ = σ(µ, c−1µ) yields the wreath recursion x = σ(yx, 1), y = (x, 1), where x = aµ and y = cµ. Further, we have x = σ(yx, 1), yx = σ(yx, x), and the last wreath recursion coincides with the one defining the automa- ton 2852. Therefore G849 ∼= G2852 (see G2852 for more information on this group). 851 ∼= G847 ∼= D4. Wreath recursion: a = σ(b, b), b=(b,b), c = (b, a). Direct calculation. 852. Basilica group B = IMG(z2 − 1). Wreath recursion: a = σ(c, b), b = (b, b), c = (b, a). This group was studied in [GŻ02a], where it is shown that B is not a sub-exponentially amenable group, it does not contain free subgroups of rank 2, and that the monoid generated by a and b is free. Some spectral considerations are provided in [GŻ02b]. Bartholdi and Virág showed in [BV05] that B is amenable, distinguishing the Basilica group as the first example of an amenable group that is not sub-exponentially amenable. 855 ∼= G847 ∼= D4. Wreath recursion: a = σ(c, c), b=(b,b), c = (b, a). Direct calculation. 856 ∼= C2 ⋉G2850. Wreath recursion: a = σ(a, a), b = (c, b), c = (b, a). All generators have order 2, hence H = 〈ba, ca〉 is normal in G856. Furthermore, ba = σ(ba, ca), ca = σ(1, ba), and therefore H = G2850. Thus G856 = 〈a〉 ⋉ H ∼= C2 ⋉ G2850, where (ba)a = (ba)−1 and (ca)a = (ca)−1. The group is not contracting since G2850 is not contracting. 857. Wreath recursion: a = σ(b, a), b = (c, b), c = (b, a). By using the approach used for G875, we can show that the forward orbit of 10∞ under the action of a is infinite, and therefore a has infinite order. Since c = (b, a) and b = (c, b), both b and c have infinite order and G857 is not a contracting group. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 858. Wreath recursion: a = σ(c, a), b = (c, b), c = (b, a). The element ab−1 = σ(1, ab−1) is the adding machine. By using the approach used for G875, we can show that the forward orbit of 10∞ under the action of a is infinite, and therefore a has infinite order. Since c = (b, a) and b = (c, b), both b and c have infinite order and G857 is not a contracting group. We have c−1b−1aba−1b = (1, a−1b−1aca−1b), a−1c−1b−1aba−1ba = (a−2b−1aca−1ba, 1), hence by Lemma 4 the group is not free. 860. Wreath recursion: a = σ(b, b), b = (c, b), c = (b, a). The element (ba−1)2 stabilizes the vertex 11 and its section at this vertex is equal to (ba−1)−1. Hence, ba−1 has infinite order. Furthermore, bc−1 = (cb−1, ba−1) implies that the order of bc−1 is infinite. Since this element stabilizes vertex 00 and its section at this vertex is equal to bc−1, all its powers belong to the nucleus. Thus, G860 is not contracting. 861. Wreath recursion: a = σ(b, b), b = (a, a), c = (b, a). The element a−1c = σ(1, c−1a) is conjugate to the adding machine and has infinite order. 864. Wreath recursion: a = σ(c, c), b = (c, b), c = (b, a). The element (ab−1)2 stabilizes the vertex 11 and its section at this vertex is equal to ab−1. Hence, ab−1 has infinite order. Furthermore, cb−1 = (bc−1, ab−1) implies that the order of cb−1 is infinite. Since this element stabilizes vertex 00 and its section at this vertex is equal to cb−1, G864 is not contracting. 865 ∼= G820 ∼= D∞. Wreath recursion: a = σ(a, a), b = (a, c), c = (b, a). All generators have order 2. Since abac = (acab, 1) and acab = (1, abac), we see that c = aba and G865 = 〈a, b〉. The section of (ba)2 at the vertex 0 is (ba)−1, so ba has infinite order and G865 ∼= D∞. Note that the group is conjugate to G932 by the automorphism δ = (aδ, δ). 866. Wreath recursion: a = σ(b, a), b = (a, c), c = (b, a). The element (c−1b)2 stabilizes the vertex 00 and its section at this vertex is equal to c−1b, which is nontrivial. Hence, c−1b has infinite order. The element (b−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to b−1a. Hence, b−1a has infinite order. Since b−1c−1ba−1ba ∣ ∣ 10 = (b−1a)b and vertex 10 is fixed under the action of b−1c−1ba−1ba we obtain that b−1c−1ba−1ba also has infinite order. Fi- nally, b−1c−1ba−1ba stabilizes the vertex 00 and has itself as a section at this vertex. Therefore G866 is not contracting. 869. Wreath recursion: a = σ(b, b), b = (a, c), c = (b, a). 128 Classification of groups generated by automata All generators have order 2. By Lemma 1 ab has infinite order, which implies that babcba also has infinite order, because it fixes the vertex 000 and its section at this vertex is equal to ab. But babcba fixed 10 and has itself as a section at this vertex. Thus, G869 is not contracting. 870: Baumslag-Solitar group BS(1, 3). Wreath recursion: a = σ(c, b), b = (a, c), c = (b, a). The automaton satisfies the conditions of Lemma 1. In particular ab has infinite order. Since bc = (ab, ca), a2 = (bc, cb), we obtain that bc and a have infinite order. Since b = (a, c), b also has infinite order. Since b has infinite order, fixes the vertex 10 and has itself as a section at this vertex, G870 is not contracting. The element µ = b−1a = σ(1, a−1b) = σ(1, µ−1) is conjugate to the adding machine and therefore has infinite order. Since a−1c = σ(1, c−1a) we see that a−1c = µ. Therefore c = ab−1a and G870 = 〈a, b〉 = 〈µ, b〉. We claim that b−1µb = µ3. Since c = ab−1a, we have ab−1ab−1ab−1a−1b = (ba−1bc−1b−1a, ca−1ba−1) = (ba−1ba−1ba−1b−1a, 1). But ba−1ba−1ba−1b−1a is a conjugate of the inverse of ab−1ab−1ab−1a−1b, which shows that ab−1ab−1ab−1a−1b = 1, and the last relation is equiva- lent to b−1µb = µ3. Since b and µ have infinite order, G870 ∼= BS(1, 3). See [BŠ06] for realizations of BS(1,m) for any value of m, m 6= ±1. 874 ∼= C2 ⋉G2852. Wreath recursion: a = σ(a, a), b = (b, c), c = (b, a). All the generators have order 2, hence H = 〈ba, ca〉 is normal in G874. Furthermore, ba = σ(ca, ba), ca = σ(1, ba), therefore H = G2852. Thus G874 = 〈a〉 ⋉H ∼= C2 ⋉G2852, where (ba)a = (ba)−1 and (ca)a = (ca)−1. In particular, G874 is not contracting and has exponential growth. 875. Wreath recursion: a = σ(b, a), b = (b, c), c = (b, a). The equalities a(10∞) = 010∞, b(10∞) = 10∞, c(10∞) = 110∞, show that all members of the forward orbit of 10∞ under the action of a have only finitely many 1’s and that the position of the rightmost 1 cannot decrease under the action of a. Since a(10∞) = 010∞, the forward orbit of 10∞ under the action of a can never return to 10∞ and a has infinite order. Note that the above equalities also show that no nonempty words w over {a, b, c} satisfies a relation of the form w = 1 in G875. First note that c = (b, a) and b = (b, c), implying that b and c have infinite order. Thus bn 6= 1, for n > 0. On the other hand, for any word w that contains I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, a or c, w(10∞) 6= 10∞ (again, since the position of the rightmost 1 moves to the right and never decreases). Since b has infinite order and b = (b, c), G875 is not contracting. 876. Wreath recursion: a = σ(c, a), b = (b, c), c = (b, a). By Lemma 2 the elements ba and acb2a2cb act transitively on the levels of the tree and, hence, have infinite order. Since (b8) ∣ ∣ 1100001100 = acb2a2cb and vertex 1100001100 is fixed under the action of b8 we obtain that b also has infinite order. Finally, b stabilizes the vertex 0 and has itself as a section at this vertex. Therefore G876 is not contracting. We have c−1b = (1, a−1c), ac−1ba−1 = (ca−1, 1), hence by Lemma 4 the group is not free. 878 ∼= C2 ⋉ IMG(1 − 1 z2 ). Wreath recursion: a = σ(b, b), b = (b, c), c = (b, a). Let x = bc and y = ca. Since all generators have order 2, the index of the subgroup H = 〈x, y〉 in G878 is 2, H is normal and G878 ∼= C2 ⋉ H, where C2 is generated by c. The action of C2 on H is given by xc = x−1 and yc = y−1. We have x = bc = (1, ca) = (1, y) and y = ca = σ(ab, 1) = σ(y−1x−1, 1). An isomorphic copy of H is obtained by exchanging the letters 0 and 1, yielding the wreath recursion x = (y, 1) and y = σ(1, y−1x−1). The last recursion defines IMG(1 − 1 z2 ) [BN06]. Thus, G878 ∼= C2 ⋉ IMG(1 − 1 z2 ). 879. Wreath recursion: a = σ(c, b), b = (b, c), c = (b, a). The element c−1a = σ(a−1c, 1) is conjugate to the adding machine and has infinite order. By Lemma 2 the element ca acts transitively on the levels of the tree and, hence, has infinite order. Since (b2) ∣ ∣ 1101 = ca and vertex 1101 is fixed under the action of b2 we obtain that b also has infinite order. Finally, b stabilizes the vertex 0 and has itself as a section at this vertex. Therefore G879 is not contracting. 882. Wreath recursion: a = σ(c, c), b = (b, c), c = (b, a). The element (ca−1cb−1)2 stabilizes the vertex 00 and its section at this vertex is equal to ca−1cb−1. Hence, ca−1cb−1 has infinite order. 883 ∼= C2 ⋉G2841. Wreath recursion: a = σ(a, a), b = (c, c), c = (b, a). All generators have order 2, hence H = 〈ba, ca〉 is normal in G883. Furthermore, ba = σ(ca, ca), ca = σ(1, ba), therefore H = G2841. Thus G883 = 〈a〉 ⋉H ∼= C2 ⋉G2841, where (ba)a = (ba)−1 and (ca)a = (ca)−1. In particular, G883 is not contracting and has exponential growth. 884. Wreath recursion: a = σ(b, a), b = (c, c), c = (b, a). The element (b−1ca−1c)2 stabilizes the vertex 0 and its section at this vertex is equal to (b−1ca−1c)−1. Hence, b−1ca−1c has infinite order. Since [b, a]2 ∣ ∣ 0100 = (b−1ca−1c)c and 0100 is fixed under the action of [b, a]2 we obtain that [b, a] also has infinite order. Finally, [b, a] stabilizes the vertex 130 Classification of groups generated by automata 00 and its section at this vertex is [b, c] = [b, a]. Therefore G884 is not contracting. 885. Wreath recursion: a = σ(c, a), b = (c, c), c = (b, a). The element (c−1b)2 stabilizes the vertex 10 and its section at this vertex is equal to c−1b. Hence, c−1b has infinite order. Furthermore, c−1b stabilizes the vertex 00 and has itself as a section at this vertex. Therefore G885 is not contracting. We have b−1aba−1 = (1, c−1aca−1), a−1b−1ab = (a−1c−1ac, 1), hence by Lemma 4 the group is not free. 887. Wreath recursion: a = σ(b, b), b = (c, c), c = (b, a). The element (ac−1)4 stabilizes the vertex 001 and its section at this vertex is equal to (ac−1)2, which is nontrivial. Hence, ac−1 has infinite order. 888. Wreath recursion: a = σ(c, b), b = (c, c), c = (b, a). The element a−1c = σ(1, c−1a) is conjugate to the adding machine and has infinite order. Since c−1b ∣ ∣ 1 = a−1c and vertex 1 is fixed under the action of c−1b we obtain that c−1b also has infinite order. Finally, c−1b stabilizes the vertex 00 and has itself as a section at this vertex. Therefore G888 is not contracting. We have c−1ab−1a = (1, a−1b), ac−1ab−1 = (ca−1bc−1, 1), hence by Lemma 4 the group is not free. 891 ∼= C2 ⋉ (Z ≀C2). Wreath recursion: a = σ(c, c), b = (c, c), c = (b, a). Let x = ac and y = cb. Since all generators have order 2, the index of the subgroup H = 〈x, y〉 in G891 is 2, H is normal and G891 ∼= C2 ⋉ H, where C2 is generated by c. The action of C2 on H is given by xc = x−1 and yc = y−1. In fact, to support the claim that H has index 2 in G891 we need to prove that c 6∈ H. We will prove a little bit more than that. Let w = 1 be a relation in G891 where w is a word over {a, b, c}. The number of occurrences of a in w must be even (otherwise w would act nontrivially on level 1). Similarly, the number of occurrences of c in w is even. Indeed, if it were odd, then exactly one of the words w0 and w1 in the decomposition w = (w0, w1) would have odd number of occurrences of the letter a, and the action of w would be nontrivial on level 2. Finally, we claim that the number of occurrences of b in w is also even. Otherwise the number of c’s in both w0 and w1 would be odd and the action of w would be nontrivial on level 3. Thus every word over {a, b, c} representing 1 must have even number of occurrences of each of the three letters. Note that this implies that the abelianization of G891 is C2 × C2 × C2. We now prove that H is isomorphic to the Lamplighter group Z ≀C2. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, The group H is self-similar, which can be seen from x = ac = σ(cb, ca) = σ(y, x−1), y = cb = (bc, ac) = (y−1, x). Consider the elements sn = σyn = y−nxyn+1, n ∈ Z (note that xy = σ). For n > 0, we have s0s1 · · · sn−1 = xnyn and s−ns−n+1 · · · s−1 = ynxn. On the other hand, sn = y−nσyn = σ(x−ny−n, ynxn) and s−n = ynσy−n = σ(xnyn, y−nx−n), implying sn = σ(s−1s−2 · · · s−n, s−n · · · s−2s−1) and s−n = σ(s0s1 · · · sn−1, sn−1 · · · s1s0). By induction on n we obtain that the depth of sn is 2n+ 1 for n ≥ 0 and the depth of s−n is 2n for n > 0 (depth of a finitary element is the lowest level at which all sections of the element are trivial). This implies that all si, i ∈ Z are different, have order 2 (they are conjugates of σ), and commute (for each i and each level m all sections of si at level m are equal). Therefore y has infinite order and H = 〈x, y〉 = 〈y, σ〉 ∼= Z ≀ C2. Since y has infinite order, stabilizes the vertex 00 and has itself as a section at this vertex, G891 is not contracting. 919 ∼= G820 ∼= D∞. Wreath recursion: a = σ(a, a), b = (a, b), c = (c, a). The states a, b form a 2-state automaton generating D∞ (see Theo- rem 7) and c = aba. 920. Wreath recursion: a = σ(b, a), b = (a, b), c = (c, a). The element (ac−1)2 stabilizes the vertex 00 and its section at this vertex is equal to ac−1. Hence, ba−1 has infinite order. 923. Wreath recursion: a = σ(b, b), b = (a, b), c = (c, a). The states a and b form a 2-state automaton generating D∞ (see Theorem 7). 924 ∼= G870. Baumslag-Solitar group BS(1, 3). Wreath recursion: a = σ(c, b), b = (a, b), c = (c, a). This fact is proved in [BŠ06]. 928 ∼= G820 ∼= D∞. Wreath recursion: a = σ(a, a), b = (b, b), c = (c, a). The states a and c form a 2-state automaton generating D∞ (see Theorem 7) and b is trivial. 929 ∼= G2851. Wreath recursion: a = σ(b, a), b = (b, b), c = (c, a). SeeG2851 for an isomorphism (in fact the groups coincide as subgroups of Aut(X∗)). 930 ∼= G821. Lamplighter group Z ≀ C2. Wreath recursion: a = σ(c, a), b = (b, b), c = (c, a). The states a and c form a 2-state automaton generating the Lamp- lighter group (see Theorem 7) and b is trivial. 132 Classification of groups generated by automata 932 ∼= G820 ∼= D∞. Wreath recursion: a = σ(b, b), b = (b, b), c = (c, a). We have b = 1 and a2 = c2 = 1. The element ac = σ(c, a) is clearly nontrivial. Since (ac)2 = (ac, ca), this element has infinite order. Thus G ∼= D∞. 933 ∼= G849. Wreath recursion: a = σ(c, b), b = (b, b), c = (c, a). See G2852 for an isomorphism between G933 and G2852 and G849 for an isomorphism between G2852 and G849. 936 ∼= G820 ∼= D∞. Wreath recursion: a = σ(c, c), b = (b, b), c = (c, a). The states a and c form a 2-state automaton generating D∞ (see Theorem 7) and b is trivial. 937 ∼= C2 ⋉G929. Wreath recursion: a = σ(a, a), b = (c, b), c = (c, a). All generators have order 2, hence H = 〈ca, ba〉 = 〈ca, caba〉 is normal in G937. Furthermore, ca = σ(1, ca), caba = σ(caba, ca), therefore H = G929. Thus G937 = 〈a〉 ⋉ H ∼= C2 ⋉ G929, where (ba)a = (ba)−1 and (ca)a = (ca)−1. In particular, G937 is regular weakly branch over H ′, has exponential growth and is not contracting. 938. Wreath recursion: a = σ(b, a), b = (c, b), c = (c, a). The element (b−1a−1ca)2 stabilizes the vertex 00 and its section at this vertex is equal to ( (b−1a−1ca)−1 )a−1c . Hence, b−1a−1ca has infinite order. Furthermore, b−1a−1ca stabilizes the vertex 1 and has itself as a section at this vertex. Therefore G938 is not contracting. We have c−1b = (1, a−1b), a−1c−1ba = (a−2ba, 1), hence by Lemma 4 the group is not free. 939. Wreath recursion: a = σ(c, a), b = (c, b), c = (c, a). The states a and c form a 2-state automaton generating the Lamp- lighter group (see Theorem 7). Hence, G939 is neither torsion, nor con- tracting, and has exponential growth. 941. Wreath recursion: a = σ(b, b), b = (c, b), c = (c, a). The second iteration of the wreath recursion is a = (02)(13)(c, b, c, b), b = (c, a, c, b), c = (23)(c, a, b, b). Conjugation by g = (cg, g, g, bg) gives the wreath recursion a′ = (02)(13), b = (c′, a′, c′, b′), c = (23)(c′, a′, 1, 1), where a′ = ag, b′ = bg, and c′ = cg. The last recursion coincides with the second iteration of the recursion α = σ, β = (γ, β), γ = (γ, α). Conjugating the last recursion by h = (γh, h) yields the recursion defining G945. Thus, G941 ∼= G945 ∼= C2 ⋉ IMG(z2 − 1) (see G945). The limit space is half of the Basilica. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, 942. Wreath recursion: a = σ(c, b), b = (c, b), c = (c, a). The Lamplighter group L = Z ≀ C2 can be defined as the group gen- erated by a′ and b′ given by the wreath recursion (see Theorem 7) a′ = σ(a′, b′), b′ = (a′, b′). Let H = 〈a, b〉 ≤ G942. We will show that H and L are isomorphic. Let Y ∗ be the subtree of X∗ consisting of all words over the alphabet Y = {01, 11}. The element b fixes the letter in Y , while a swaps them. Since a01 = b01 = a, a11 = b11 = b, the tree Y ∗ is invariant under the action of H. Moreover, the action of H on Y ∗ coincides with the action of the Lamplighter group L = 〈a′, b′〉 on X∗ (after the identification 01 ↔ 0, 11 ↔ 1). This implies that the map φ : H → L given by a 7→ a′, b 7→ b′ can be extended to a homomorphism. We claim that this homomorphism is in fact an isomorphism. Let w = w(a, b) be a group word representing an element of the kernel of φ. Since w(a′, b′) represents the identity in the lamplighter group L, the total exponent of a in w must be even and the total exponent ε of both a and b in w must be 0. Therefore the element g = w(a, b) stabilizes the top two levels of the tree X∗ and can be decomposed as g = (cε, ∗, cε, ∗), where the ∗’s are words over a and b representing the identity in H (these words correspond precisely to the first level sections of w(a′, b′) in L). Since ε = 0, we see that g = 1 and the kernel of φ is trivial. Thus, the Lamplighter group is a subgroup of G942, which shows that G942 is not a torsion group, it is not free, and has exponential growth. Since b = (c, b) and b has infinite order, G942 is not a contracting group. 945 ∼= G941 ∼= C2 ⋉ IMG(z2 − 1). Wreath recursion: a = σ(c, c), b = (c, b), c = (c, a). All generators have order 2. Since ab = σ(1, cb) and cb = (1, ab) we see that H = 〈ab, cb〉 ∼= G852 = IMG(z2 − 1). This subgroup is normal in G945 because the generators have order 2. Since G945 = 〈H, b〉, it has a structure of a semidirect product 〈b〉⋉H = C2 ⋉IMG(z2−1) with the action of b on H given by (ab)b = (ab)−1 and (cb)b = (cb)−1. It follows that G945 is regular weakly branch over H ′ and has exponential growth. See G941 for an isomorphism. 955 ∼= G937 ∼= C2 ⋉ G929. Wreath recursion: a = σ(a, a), b = (b, c), c = (c, a). All generators have order 2. Consider the subgroup H = 〈ba = σ(ca, ba), ca = σ(1, ca)〉 ∼= G929. This subgroup is normal in G955 be- cause all generators have order 2. Since G955 = 〈H, a〉, it has a structure 134 Classification of groups generated by automata of a semidirect product 〈a〉 ⋉H = C2 ⋉G929 with the action of a on H given by (ba)b = (ba)−1 and (ca)b = (ca)−1. It is proved above that G937 has the same structure. It follows that G955 is regular weakly branch over H ′ and has exponential growth. 956. Wreath recursion: a = σ(b, a), b = (b, c), c = (c, a). The element (c−1b)2 stabilizes the vertex 10 and its section at this vertex is equal to (c−1b)−1. Hence, c−1b has infinite order. Furthermore, c−1b stabilizes the vertex 0 and has itself as a section at this vertex. Therefore G956 is not contracting. We have c−1b−1aba−1b = (1, a−1c−1aba−1c), a−1c−1b−1aba−1ba = (a−2c−1aba−1ca, 1), hence by Lemma 4 the group is not free. 957. Wreath recursion: a = σ(c, a), b = (b, c), c = (c, a). The states a, c form a 2-state automaton generating the Lamplighter group (see Theorem 7). Hence, G957 is neither torsion, nor contracting and has exponential growth. 959. Wreath recursion: a = σ(b, b), b = (b, c), c = (c, a). The element (a−1c)4 stabilizes the vertex 00 and its section at this vertex is equal to (a−1c)−1. Hence, a−1c has infinite order. Furthermore, since c−1b = (c−1b, a−1c), this element also has infinite order. Thus, G959 is not contracting. 960. Wreath recursion: a = σ(c, b), b = (b, c), c = (c, a). Define x = ac−1, y = ba−1 and z = cb−1. Then x = σ(1, y), y = σ(z, z−1) and z = (z, x). The element (zxy)8 stabilizes the vertex 001010 and its section at this vertex is equal to xy−1z = xyz = (zxy)z−1 (since y2 = 1). Hence, zxy has infinite order. Denote t = (b−1c)4(b−1a)(c−1a)5(b−1c). Then t2 stabilizes the vertex 00 and t2 ∣ ∣ 00 = tb −1c. Hence, t has infinite order. Let s = c−2b2. Since s32 ∣ ∣ 111000000100 = tc and s32 fixes 111000000100, we obtain that s also has infinite order. Finally, s stabilizes the vertex 00 and has itself as a section at this vertex. Therefore G960 is not contracting. 963. Wreath recursion: a = σ(c, c), b = (b, c), c = (c, a). All generators have order 2. The element ac = σ(1, ca) is conjugate to the adding machine and has infinite order. Furthermore, since cb = (cb, ac), this element also has infinite order. Thus, G963 is not contracting. 964 ∼= G739 ∼= C2 ⋉ (C2 ≀ Z). Wreath recursion: a = σ(a, a), b = (c, c), c = (c, a). All generators have order 2. The elements u = acba = (ca, 1) and v = bc = (1, ca) generate Z 2 because ca = σ(1, ca) is the adding machine and has infinite order. We have cacb = σ and 〈u, v〉 is normal in H = 〈u, v, σ〉 I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, because uσ = v and vσ = u. In other words, H ∼= C2 ⋉ (Z × Z) = C2 ≀ Z. Furthermore, G964 = 〈H, a〉 and H is normal in G972 because ua = v−1, va = u−1 and σa = σ. Thus G964 = C2⋉(C2 ≀Z), where the action of C2 on H is specified above and coincides with the one in G739. Therefore G964 ∼= G739. 965. Wreath recursion: a = σ(b, a), b = (c, c), c = (c, a). The element (ac−1)2 stabilizes the vertex 01 and its section at this vertex is equal to (ac−1)−1. Hence, ac−1 has infinite order. By Lemma 2 the element a acts transitively on the levels of the tree and, hence, has infinite order. Since c = (c, a) we obtain that c also has infinite order. Therefore G965 is not contracting. We have bc−1 = (1, ca−1), a−1bc−1a = (a−1c, 1), hence by Lemma 4 the group is not free. 966. Wreath recursion: a = σ(c, a), b = (c, c), c = (c, a). The states a and c form a 2-state automaton generating the Lamp- lighter group (see Theorem 7). Hence, G966 is neither torsion, nor con- tracting, and has exponential growth. Since b = (c, c) we obtain that G966 can be embedded into the wreath product C2 ≀ (Z ≀ C2). This shows that G966 is solvable. 968. Wreath recursion: a = σ(b, b), b = (c, c), c = (c, a). We will show that this group contains Z 5 as a subgroup of index 16. It is a contracting group, with nucleus consisting of 73 elements (the self-similar closure of the nucleus consists of 77 elements). All generators have order 2. Let x = (ac)2, y = bcba, and K = 〈x, y〉. Conjugating x and y by γ = (bγ, aγ) yields the self-similar copy K ′ of K generated by x′ = ((y′)−1, (y′)−1) and y = σ(x′, y′), where x′ = xγ and y′ = yγ . Since [x′, y′] = ([x′, y′](y ′)−1 , 1) K ′ is abelian. The matrix of the corresponding virtual endomorphism is given by A = ( 0 1 2 −1 1 2 ) . The eigenvalues λ = 1 4 ± 1 4 √ 7i of this matrix are not algebraic integers. Therefore K ′ (ad therefore K as well) is free abelian of rank 2, by the results in [NS04]. The subgroup H = 〈ab, bc〉 has index 2 in G968 (the generators of G968 have order 2). The second level stabilizer StabH(2) has index 8 in H (the quotient group is isomorphic to the dihedral group D4). The stabilizer StabH(2), is generated by (bc)2, ( (bc)2 )ba , (ab)2, ( (ab)2 )bc , ( (ab)2 )(bc)ba , 136 Classification of groups generated by automata and ( (ab)2 )bc(bc)ba . Conjugating these elements by g = (b, c, b, 1) gives g1 = ( (bc)2 )g = (bcbc)g = (1, 1, y, y−1 ), g2 = ( (bc)2 )bag = (acbcba)g = (y, y, 1, 1 ), g3 = ( (ab)2 )bcg = (cbabac)g = (1, x, x, 1 ), g4 = ( (ab)2 )g = (abab)g = (1, x, 1, x−1 ), g5 = ( (ab)2 )(bc)bag = (abcbabacba)g = (x, 1, 1, x−1 ), g6 = ( (ab)2 )bc(bc)bag = (abcacbabacacba)g = (x, 1, x, 1 ). Therefore, StabH(2) is abelian and g6 = g5g3g −1 4 . If ∏5 i=1 g ni i = 1, then xn5yn2 = xn3+n4yn2 = xn3yn1 = xn4+n5yn1 = 1. Since K is free abelian, we obtain ni = 0, i = 1, . . . , 5. Therefore StabH(2) is a free abelian group of rank 5. 969. Wreath recursion: a = σ(c, b), b = (c, c), c = (c, a). The element (cb−1)4 stabilizes the vertex 100 and its section at this vertex is equal to cb−1. Hence, cb−1 has infinite order. We have bc−1 = (1, ca−1), ca−1 = σ(ab−1, 1), ab−1 = σ(1, bc−1), hence the subgroup generated by these elements is isomorphic to IMG(1 − 1 z2 ) (see [BN06]). We also have c−1b = (1, a−1c), a−1c−1ba = (b−1a−1cb, 1), hence by Lemma 4 the group is not free. 972 ∼= G739 ∼= C2 ⋉ (C2 ≀ Z). Wreath recursion : a = σ(c, c), b = (c, c), c = (c, a). All generators have order 2. The elements u = acba = (ca, 1) and v = bc = (1, ac) generate Z 2 because ca = σ(ac, 1) is conjugate to the adding machine and has infinite order. Also we have ba = σ and 〈u, v〉 is normal in H = 〈u, v, σ〉 because uσ = v and vσ = u. In other words, H ∼= C2 ⋉ (Z × Z) = C2 ≀ Z. Furthermore, G972 = 〈H, a〉 and H is normal in G972 because ua = v−1, va = u−1 and σa = σ. Thus G972 = C2⋉(C2 ≀Z), where the action of C2 on H is specified above and coincides with the one in G739. Therefore G972 ∼= G739. 1090 ∼= C2. Wreath recursion: a = σ(a, a), b = (b, b), c = (b, b). Both b and c are trivial and a2 = 1. 1091 ∼= G731 ∼= Z. Wreath recursion: a = σ(b, a), b = (b, b), c = (b, b). Both b and c are trivial and a is the adding machine. 1094 ∼= G1090 ∼= C2. Wreath recursion: a = σ(b, b), b = (b, b), c = (b, b). Both b and c are trivial and a2 = 1. 2190 ∼= G848 ∼= C2 ≀ Z. Wreath recursion: a = σ(c, a), b = σ(a, a), c = (a, a). First note that c = a−2. Therefore G = 〈a, b〉, where a = σ(a−2, a), and b = σ(a, a). Also, a has infinite order. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Consider the subgroup H = 〈ba, ab〉 < G. The generators of H commute since ba = (a−1, a2) and ab = (a2, a−1). Furthermore, (ba)n(ab)m = (a−n+2m, a2n−m) = 1 if and only if m = n = 0. Therefore H ∼= Z 2. Consider the element ba2 = bc−1 = σ. This element does not belong to H, since H stabilizes the first level of the tree. On the other hand a = (ba)−1ba2 = (ba)−1σ and b = a−1(ab) so G = 〈σ,H〉. Finally, (ba)σ = ab and (ab)σ = ba implies that H is normal in G and G = C2 ≀H ∼= C2 ≀ Z ∼= G848. Also note that 〈a, ab〉 = G2212 ∼= Z ∗2Z Z. 2193. Wreath recursion: a = σ(c, b), b = σ(a, a), c = (a, a). Let x = ca−1 and y = ab−1. Then x = σ(ab−1, ac−1) = σ(y, x−1) and y = (ba−1, ca−1) = (y−1, x). It is already shown (see G891), that 〈x, y〉 is not contracting and is isomorphic to the Lamplighter group. Therefore G2193 is not a torsion group, it is not contracting, and has exponential growth. 2196 ∼= G802 ∼= C2 ×C2 ×C2. Wreath recursion: a = σ(c, c), b = σ(a, a), c = (a, a). Direct calculation. 2199. Wreath recursion: a = σ(c, a), b = σ(b, a), c = (a, a). By Lemma 2 the element ac acts transitively on the levels of the tree and, hence, has infinite order. Since ba = (ac, ba) we obtain that ba also has infinite order. Therefore G2199 is not contracting. We have b−2abcba = b−2aba−2ba = 1, and a and b do not commute, hence the group is not free. 2202. Wreath recursion: a = σ(c, b), b = σ(b, a), c = (a, a). The element (b−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to b−1a. Hence, b−1a has infinite order. Furthermore, b−1a stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2202 is not contracting. We have cb−1c−1b = (1, ab−1a−1b), bcb−1c−1 = (bab−1a−1, 1), hence by Lemma 4 the group is not free. 2203. Wreath recursion: a = σ(a, c), b = σ(b, a), c = (a, a). The states a and c form a 2-state automaton generating the infinite cyclic group Z in which c = a−2 (see Theorem 7). Since b−1a ∣ ∣ 1 = a−1c and vertex 1 is fixed under the action of b−1a we obtain that b−1a also has infinite order. Finally, b−1a stabilizes the vertex 0 and has itself as a section at this vertex. Therefore G2203 is not contracting. We have c−2ab = (1, a−2cb), bc−2a = (ba−2c, 1), hence by Lemma 4 the group is not free. 2204. Wreath recursion: a = σ(b, c), b = σ(b, a), c = (a, a). 138 Classification of groups generated by automata The element (b−1ac−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to b−1ac−1a. Hence, b−1ac−1a has infinite order. Since [c, a]2 ∣ ∣ 000 = (b−1ac−1a)a−1cb and 000 is fixed under the action of [c, a]2 we obtain that [c, a] also has infinite order. Finally, [c, a] stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2204 is not contracting. We have ab−1 = (1, ca−1), b−1a = (a−1c, 1), hence by Lemma 4 the group is not free. 2205 ∼= G775 ∼= C2 ⋉ IMG ( ( z−1 z+1 )2 ) . Wreath recursion: a = σ(c, c), b = σ(b, a), c = (a, a). See G783 for an isomorphism between G783 and G2205. 2206 ∼= G748 ∼= D4 × C2. Wreath recursion: a = σ(a, a), b = σ(c, a), c = (a, a). Direct calculation. 2207. Wreath recursion: a = σ(b, a), b = σ(c, a), c = (a, a). The element (c−1a)4 stabilizes the vertex 000 and its section at this vertex is equal to c−1a. Hence, c−1a has infinite order. Since b−1a−1b−1aba ∣ ∣ 001 = (c−1a)a and the vertex 001 is fixed under the action of b−1a−1b−1aba we obtain that b−1a−1b−1aba also has infinite order. Finally, b−1a−1b−1aba stabilizes the vertex 000 and has itself as a section at this vertex. Therefore G2207 is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2209. Wreath recursion: a = σ(a, b), b = σ(c, a), c = (a, a). The element (b−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to (b−1a)−1. Hence, b−1a has infinite order. Furthermore, b−1a stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2209 is not contracting. We have aca−2c−1acac−1a−2cac−1 = 1, and a and c do not commute, hence the group is not free. 2210. Wreath recursion: a = σ(b, b), b = σ(c, a), c = (a, a). The element (a−1c)2 stabilizes the vertex 000 and its section at this vertex is equal to a−1c. Hence, a−1c has infinite order. Since (b−1a)2 ∣ ∣ 00 = a−1c and 00 is fixed under the action of b−1a we obtain that b−1a also has infinite order. Finally, b−1a stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2210 is not contracting. We have c−1b−1cb = (1, a−1c−1ac), bc−1b−1c = (ca−1c−1a, 1), hence by Lemma 4 the group is not free. 2212. Klein bottle group, 〈a, b ∣ ∣ a2 = b2〉. Wreath recursion: a = σ(a, c), b = σ(c, a), c = (a, a). I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, The states a and c form a 2-state automaton generating the infinite cyclic group Z in which c = a−2 (see Theorem 7). We have a = σ(a, a−2), b = σ(a−2, a), and x = ab−1 = (a−3, a3). Finally, since xa = b−1a = (a3, a−3) = x−1, we have G2212 = 〈x, a \| xa = x−1〉 and G2212 is the Klein bottle group. Tietze transformations yield the presentation G2212 = 〈a, b \| a2 = b2〉 in terms of the generators a and b. 2213. Wreath recursion: a = σ(b, c), b = σ(c, a), c = (a, a). By Lemma 2 the element cb acts transitively on the levels of the tree and, hence, has infinite order. Since (ba) ∣ ∣ 100 = cb and the vertex 100 is fixed under the action of ba we obtain that ba also has infinite order. Finally, ba stabilizes the vertex 01 and has itself as a section at this vertex. Therefore G2213 is not contracting. We have c−1b−1cb = (1, a−1c−1ac), bc−1b−1c = (ca−1c−1a, 1), hence by Lemma 4 the group is not free. 2214 ∼= G748 ∼= D4 × C2. Wreath recursion: a = σ(c, c), b = σ(c, a), c = (a, a). Direct calculation. 2226 ∼= G820 ∼= D∞. Wreath recursion: a = σ(c, a), b = σ(b, b), and c = (a, a). We have ba = (bc, ba), bc = σ(ba, ba), and b = σ(b, b). Therefore x, y and b satisfy the wreath recursion defining the automaton A2394. Thus G2226 = G2394 ∼= G820. 2229. Wreath recursion: a = σ(c, b), b = σ(b, b), c = (a, a). Note that b is of order 2. Post-conjugating the recursion by (1, b) (which is equivalent to conjugating by the tree automorphism g = (g, bg) in Aut(X∗) gives a copy of G2229 defined by a = σ(bc, 1), b = σ, c = (a, bab) The stabilizer of the first level is generated by a2 = (bc, bc), c = (a, bab), ba = (bc, 1), bcb = (bab, a). Its projection on the first level is generated by bc = σ(a, bab), a = σ(bc, 1), bab = σ(1, bc). Furthermore, bcbc = (baba, abab), abab = (1, bcbc), baba = (bcbc, 1), which implies that bc is of order 2 and a−1 = bab. Hence, the projection of the stabilizer on the first level is generated by the recursion a = σ(bc, 1), bc = σ(a, a−1). 140 Classification of groups generated by automata Post-conjugating by (1, a), we obtain the recursion a = σ(a−1 · bc, a), bc = σ, which is the group C4 ⋉Z 2 of all orientation preserving automorphisms of the integer lattice (see [BN06]). Note that the nucleus of G2229 consists of 52 elements. 2232 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(c, c), b = σ(b, b), c = (a, a). Direct calculation. 2233. Wreath recursion: a = σ(a, a), b = σ(c, b), c = (a, a). Therefore, 〈ba = (ba, ca), ca = σ〉 = G932 ∼= D∞. Conjugating by g = (ag, g), we obtain the recursion α = σ, β = σ(γβ, αβ), γ = (α, α), where α = ag, β = bg, and γ = cg. Therefore α = σ, αβ = (γα, αβ), γα = σ(α, α), and the last wreath recursion defines a bounded automaton (see Section 3 for a definition). It follows from [BKN] that G2233 is amenable. 2234. Wreath recursion: a = σ(b, a), b = σ(c, b), c = (a, a). The element (c−1b)4 stabilizes the vertex 00 and its section at this vertex is equal to (c−1b)−1. Hence, c−1b has infinite order. Since (b−1a) ∣ ∣ 0 = c−1b and 0 is fixed under the action of b−1a we obtain that b−1a also has infinite order. Finally, b−1a stabilizes the vertex 1 and has itself as a section at this vertex. Therefore G2234 is not contracting. We have c−1b−1ac−1a2 = (1, a−1c−1b2), ac−1b−1ac−1a = (ba−1c−1b, 1), hence by Lemma 4 the group is not free. 2236. Wreath recursion: a = σ(a, b), b = σ(c, b), c = (a, a). By Lemma 2 the element b acts transitively on the levels of the tree and, hence, has infinite order. By Lemma 2 the element cb acts transitively on the levels of the tree and, hence, has infinite order. Since ba = (ba, cb) we obtain that ba also has infinite order. Since ba has itself as a section at 0 the group is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2237. Wreath recursion: a = σ(b, b), b = σ(c, b), c = (a, a). By Lemma 2 the elements b and (bc)3 acts transitively on the levels of the tree and, hence, have infinite order. Since (cba)2 ∣ ∣ 00000 = (bc)3 and 00000 is fixed under the action of (cba)2 we obtain that cba also has infinite order. Finally, cba stabilizes the I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, vertex 101 and has itself as a section at this vertex. Therefore G2237 is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2239. Wreath recursion: a = σ(a, c), b = σ(c, b), c = (a, a). The group contains elements of infinite order by Lemma 1. In par- ticular, ca has infinite order. Since (ba) ∣ ∣ 100 = ca and the vertex 100 is fixed under the action of ba we obtain that ba also has infinite order. Fi- nally, ba stabilizes the vertex 1 and has itself as a section at this vertex. Therefore G2239 is not contracting. We have ca−2cba−1 = (1, c−1abc−1), a−1ca−2cb = (c−2ab, 1), hence by Lemma 4 the group is not free. We can also simplify the wreath recursion in the following way. Since c = a−2 we have a = σ(a, a−2), b = σ(a−2, b). Therefore ab = (a−4, ab), a = σ(a, a−2), which can be written as ab = (a−4, ab), a = σ(1, a−1), which is a subgroup of β = (a, β), a = σ(1, a−1). 2240. Free group of rank 3. Wreath recursion: a = σ(b, c), b = σ(c, b), c = (a, a). The automaton appeared for the first time in [Ale83]. The fact that G2240 is free group of rank 3 with basis {a, b, c} is proved in [VV05]. This is the smallest automaton among all automata over a 2-letter alphabet generating a free nonabelian group. The fact that G2240 is not contracting follows now from the result of Nekrashevych [Nek07a], that a contracting group cannot have free subgroups. Alternatively, b−1ca has infinite order, stabilizes the vertex 11 and has itself as a section at this vertex. Hence, the group is not contracting. 2241 ∼= G739 ∼= C2 ⋉ (C2 ≀ Z). Wreath recursion: a = σ(c, c), b = σ(c, b), c = (a, a). Consider G747. Its wreath recursion is given by a = σ(c, c), b = (b, a), c = (a, a). All generators have order 2 and a commutes with c. Therefore 142 Classification of groups generated by automata acb = σ(cab, c) = σ(acb, c) and wa have G747 = 〈a, acb, c〉 = G2241. Thus G2241 = G747 ∼= G739. 2260 ∼= G802 ∼= C2 × C2 × C2. Wreath recursion: a = σ(a, a), b = (c, c), c = (a, a). Direct calculation. 2261. Wreath recursion: a = σ(b, a), b = σ(c, c), c = (a, a). The element (ac−1)2 stabilizes the vertex 00 and its section at this vertex is equal to (ac−1)−1. Hence, ac−1 and c−1a have infinite order. Since b−1c−1ac−1ba ∣ ∣ 001 = ( (c−1a)2 )a and the vertex 001 is fixed under the action of b−1c−1ac−1ba we obtain that b−1c−1ac−1ba also has infinite order. Finally, b−1c−1ac−1ba stabilizes the vertex 000 and has itself as a section at this vertex. Therefore G2261 is not contracting. We have acac−1a−2cac−1aca−2c−1 = 1, and a and c do not commute, hence the group is not free. 2262 ∼= G848 ∼= C2 ≀ Z. Wreath recursion: a = σ(c, a), b = σ(c, c), c = (a, a). The states a and c form a 2-state automaton (see Theorem 7). More- over, c = a−2 and a has infinite order. Thus a = σ(a−2, a), b = σ(a−2, a−2) and G2262 = 〈a, b〉. Further, b−1a = (1, a3) and a−3 = σ(1, a3), yielding a−4b = σ. Therefore G = 〈a, σ〉. Since 〈a, aσ〉 = Z 2, we obtain that G2262 ∼= C2 ≀ Z2 ∼= G848. 2264 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(b, b), b = σ(c, c), c = (a, a). Direct calculation. 2265. Wreath recursion: a = σ(c, b), b = σ(c, c), c = (a, a). The element (c−1b)4 stabilizes the vertex 0000 and its section at this vertex is equal to ( (c−1b)−1 )c−1a . Hence, c−1b has infinite order. Since [c, a] ∣ ∣ 10 = (c−1b)c and 10 is fixed under the action of [c, a] we obtain that [c, a] also has infinite order. Finally, [c, a] stabilizes the vertex 00 and has itself as a section at this vertex. Therefore G2265 is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2271. Wreath recursion: a = σ(c, a), b = σ(a, a), c = (b, a). The element (ac−1)4 stabilizes the vertex 001 and its section at this vertex is equal to ac−1. Hence, ac−1 has infinite order. The element (a−1b)4 stabilizes the vertex 000 and its section at this vertex is equal to a−1b. Hence, a−1b has infinite order. Since b−1c−1ac−1a2 ∣ ∣ 001 = (a−1b)a and the vertex 001 is fixed under the action of b−1c−1ac−1a2 we obtain that b−1c−1ac−1a2 also has infinite order. Fi- nally, b−1c−1ac−1a2 stabilizes the vertex 000 and has itself as a section at this vertex. Therefore G2271 is not contracting. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2274. Wreath recursion: a = σ(c, b), b = σ(a, a), c = (b, a). The element a−1c = σ(1, c−1a) is conjugate to the adding machine and has infinite order. Since (b−1a) ∣ ∣ 0 = a−1c and 0 is fixed under the action of b−1a we obtain that b−1a also has infinite order. Finally, b−1a stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2274 is not contracting. We have bc−2b = (1, ab−2a), b2c−2 = (a2b−2, 1), hence by Lemma 4 the group is not free. 2277 ∼= C2 ⋉ (Z × Z). Wreath recursion: a = σ(c, c), b = σ(a, a), c = (b, a). All generators have order 2. Let x = cb, y = ab and H = 〈x, y〉. We have x = σ(1, y−1) and y = (xy−1, xy−1). The elements x and y commute and the matrix of the associated virtual endomorphism is given by A = ( 0 1 −1/2 −1 ) . The eigenvalues −1 2 ± 1 2 i are not algebraic integers, and therefore, accord- ing to [NS04], H is free abelian of rank 2. The subgroup H is normal of index 2 in G2277. Therefore G2277 = 〈H, b〉 = C2 ⋉ (Z × Z), where C2 is generated by b, which acts on H is inversion of the generators. 2280. Wreath recursion: a = σ(c, a), b = σ(b, a), c = (b, a). We prove that a has infinite order by considering the forward orbit of 10∞ under the action of a2. We have a2 = (ac, ca), ac = σ(cb, a2), ca = σ(ac, ba) cb = σ(ab, ba), ba = (ac, ba), ab = (ab, ca). The equalities a2(10∞) = ab(10∞) = 1110∞, ac(10∞) = ca(10∞) = cb(10∞) = 0010∞, and ba(10∞) = 10110∞ show that all members of the forward orbit of 10∞ under the action of a2 have only finitely many 1’s and that the position of the rightmost 1 cannot decrease under the action of a2. Since a2(10∞) = 1110∞, the forward orbit of 10∞ under the action of a2 can never return to 10∞ and a2 has infinite order. Since a2 = (ac, ca), the elements ca and ab = (ab, ca) have infinite order, showing that G2280 is not contracting. 144 Classification of groups generated by automata 2283. Wreath recursion: a = σ(c, b), b = σ(b, a), c = (b, a). By Lemma 2 the element ac acts transitively on the levels of the tree and, hence, has infinite order. Since ba = (ac, b2) we obtain that ba also has infinite order. Finally, ba stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2283 is not contracting. 2284. Wreath recursion: a = σ(a, c), b = σ(b, a), c = (b, a). Define u = b−1a, v = a−1c and w = c−1b. Then u = (u, v), v = σ(w, 1) and w = σ(u−1, u). The group 〈u, v, w〉 is generated by the automaton symmetric to the one generating the subgroup 〈x, y, z〉 of G960 (see G960 for the definition). It is shown above that zxy has infinite order. Therefore wvu also has infinite order. The element (b−1ac−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to (b−1ac−1a)a−1b. Hence, b−1ac−1a has infinite order. Let t = b−1ab−2a2. Since t\|110 = b−1ac−1a and the vertex 110 is fixed under the action of t we see that t also has infinite order. Finally, t stabilizes the vertex 11101000 and has itself as a section at this vertex. Therefore G2284 is not contracting. 2285. Wreath recursion: a = σ(b, c), b = σ(b, a), c = (b, a). The element ac−1 = σ(1, ca−1) is conjugate to the adding machine and has infinite order. By Lemma 2 the element abcb acts transitively on the levels of the tree and, hence, has infinite order. Since (ba)2 ∣ ∣ 000 = (ac, b2) and the vertex 000 is fixed under the action of (ba)2 we obtain that ba also has infinite order. Finally, ba stabilizes the vertex 01 and has itself as a section at this vertex. Therefore G2285 is not contracting. 2286. Wreath recursion: a = σ(c, c), b = σ(b, a), c = (b, a). The element (c−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to (c−1a)a−1b. Hence, c−1a has infinite order. Since (c−2a2) ∣ ∣ 000 = (c−1a)b−1 and 000 is fixed under the action of c−2a2 we obtain that c−2a2 also has infinite order. Finally, c−2a2 stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2286 is not contracting. 2287. Wreath recursion: a = σ(a, a), b = σ(c, a), c = (b, a). The element bc−1 = σ(cb−1, 1) is conjugate to the adding machine and has infinite order. Conjugating the generators by g = (g, ag), we obtain the wreath recursion a′ = σ, b′ = σ(a′c′, 1), c′ = (b′, a′), where a′ = ag, b′ = bg, and c′ = cg. Therefore a′ = σ, b′ = σ(a′c′, 1), a′c′ = σ(b′, a′) I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, A direct computation shows that the iterated monodromy group of z2+2 1−z2 is generated by α = σ, β = σ(γ−1β−1, α), γ = (βγβ−1, α), where α, β, and γ are loops around the post-critical points 2, −1 and −2, respectively (recall the definition of iterated monodromy group in Section 5). We see that α = σ, βγ = σ(β−1, 1), β = σ(γ−1β−1, α) satisfy the same recursions as a, b and ac, only composed with taking inverses. If we take second iteration of the wreath recursions, we obtain isomorphic self-similar groups. It follows that the group G2287 is isomorphic to IMG ( z2+2 1−z2 ) and the limit space is homeomorphic to the Julia set of this rational function. 2293. Wreath recursion: a = σ(a, c), b = σ(c, a), c = (b, a). The element (b−1c)2 stabilizes the vertex 0 and its section at this vertex is equal to (b−1c)−1. Hence, b−1c has infinite order. Since (c−1bc−1a)2 ∣ ∣ 000 = b−1c and 000 is fixed under the action of (c−1bc−1a)2 we obtain that c−1bc−1a also has infinite order. Finally, c−1bc−1a stabi- lizes the vertex 11 and has itself as a section at this vertex. Therefore G2293 is not contracting. We have b−1c2a−1 = (1, c−1b2c−1), c2a−1b−1 = (b2c−2, 1), hence by Lemma 4 the group is not free. 2294. Baumslag-Solitar group BS(1,−3). Wreath recursion: a = σ(b, c), b = σ(c, a), c = (b, a). The automaton satisfies the conditions of Lemma 1. Therefore cb has infinite order. Since a2 = (cb, bc), c = (b, a) and ba = (ab, c2), the elements a, c and ba have infinite order. Finally, ba fixes the vertex 01 and has itself as a section at this vertex, showing that G2294 is not contracting. Let µ = ca−1. We have µ = ca−1 = σ(ac−1, 1) = σ(µ−1, 1), and therefore µ is conjugate of the adding machine and has infinite order. Further, we have bc−1 = σ(cb−1, 1) = σ((bc−1)−1, 1), showing that bc−1 = µ = ca−1. Therefore G2294 = 〈µ, a〉. It can be shown that aµa−1 = µ−3 in G2294 (compare to G870. Since both a and µ have infinite order G2294 ∼= BS(1,−3). 2295. Wreath recursion: a = σ(c, c), b = σ(c, a), c = (b, a). The element cb−1 = σ(1, bc−1) is conjugate to the adding machine and has infinite order. Hence, its conjugate a−1cb−1a also has infinite order. Since c−1ac−1b = ( c−1ac−1b, a−1cb−1a ) , the element c−1ac−1b has infinite order and G2295 is not contracting. 146 Classification of groups generated by automata We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2307. Contains G933. Wreath recursion: a = σ(c, a), b = σ(b, b), c = (b, a). We have ba = (bc, ba), and bc = σ(1, ba). ThereforeG933 is a subgroup of G2307 (the wreath recursion for ba and bc defines an automaton that is symmetric to the one defining the automaton [993]). The element (a−1b)2 stabilizes the vertex 00 and its section at this vertex is equal to a−1b. Hence, a−1b has infinite order. Furthermore, a−1b stabilizes the vertex 1 and has itself as a section at this vertex. Therefore G2307 is not contracting. 2313 ∼= G2277 ∼= C2 ⋉ (Z×Z). Wreath recursion: a = σ(c, c), b = σ(b, b), c = (b, a). Since all generators have order 2 the subgroup H = 〈ba, bc〉 is normal in G2313. Furthermore, ba = σ(bc, bc) and bc = σ(1, ba). Hence, H = G771 ∼= Z 2. Finally, G2313 = 〈H, b〉 = 〈b〉 ⋉ H = C2 ⋉ (Z × Z), where b inverts the generators of H. This action coincides with the one for G2277, which proves that these groups are isomorphic. 2320 ∼= G2294. Baumslag-Solitar group BS(1,−3). Wreath recursion: a = σ(a, c), b = σ(c, b), c = (b, a). It is proved in [BŠ06] that the automaton [2320] generates BS(1,−3). 2322. Wreath recursion: a = σ(c, c), b = σ(c, b), c = (b, a). The element (a−1c)2 stabilizes the vertex 00 and its section at this vertex is equal to (a−1c)b−1 . Hence, a−1c has infinite order. Since (c−2a2)2 ∣ ∣ 000 = a−1c and 000 is fixed under the action of c−2a2 we ob- tain that c−2a2 also has infinite order. Finally, c−2a2 stabilizes the ver- tex 11 and has itself as a section at this vertex. Therefore G2322 is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2352 ∼= G740. Wreath recursion: a = σ(c, a), b = σ(a, a), c = (c, a). We have ac−1b = (a, a). Therefore G2352 = 〈a, ac−1b, c〉 = G740. 2355. Wreath recursion: a = σ(c, b), b = σ(a, a), c = (c, a). The element (b−1a)2 stabilizes the vertex 00 and its section at this vertex is equal to (b−1a)a−1c. Hence, b−1a has infinite order. Further- more, b−1a stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2355 is not contracting. We have a−1cb−1c = (b−1c, 1), cb−1ca−1 = (1, cb−1), hence by Lemma 4 the group is not free. 2358 ∼= G820 ∼= D∞. Wreath recursion: a = σ(c, c), b = σ(a, a), c = (c, a). I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, The states a and c form a 2-state automaton generating D∞ (see Theorem 7) and b = aca. 2361. Wreath recursion: a = σ(c, a), b = σ(b, a), c = (c, a). The element bc−1 = σ(bc−1, 1) is conjugate to the adding machine and has infinite order. 2364. Wreath recursion: a = σ(c, b), b = σ(b, a), c = (c, a). The element cb−1 = σ(1, cb−1) is the adding machine and has infi- nite order. Therefore its conjugate b−1c also has infinite order. Since (b−1a) ∣ ∣ 0 = b−1c and 0 is fixed under the action of b−1a we obtain that b−1a also has infinite order. Finally, b−1a stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2364 is not contracting. We have c−1ac−1b = (1, a−1bc−1b), bc−1ac−1 = (ba−1bc−1, 1), hence by Lemma 4 the group is not free. 2365. Wreath recursion: a = σ(a, c), b = σ(b, a), c = (c, a). By Lemma 2 the element cb acts transitively on the levels of the tree and, hence, has infinite order. 2366. Wreath recursion: a = σ(b, c), b = σ(b, a), c = (c, a). By Lemma 2 the element a acts transitively on the levels of the tree and, hence, has infinite order. Since c = (c, a) we obtain that c also has infinite order and G2366 is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2367. Wreath recursion: a = σ(c, c), b = σ(b, a), c = (c, a). The states a and c form a 2-state automaton generating D∞ (see Theorem 7). Also we have bc = σ(bc, 1) and ca = σ(ac, 1). Therefore the elements bc and ca generate the Brunner-Sidki-Vierra group (see [BSV99]). 2368 ∼= G739 ∼= C2 ⋉ (C2 ≀Z). Wreath recursion: a = σ(a, a), b = σ(c, a), c = (c, a). We have bc−1a = (a, a). Therefore G2368 = 〈a, c, bc−1a〉 = G739. 2369. Wreath recursion: a = σ(b, a), b = σ(c, a), c = (c, a). By using the approach already used for G875, we can show that the forward orbit of 10∞ under the action of a is infinite, and therefore a has infinite order. Since a2 = (ab, ba), the element ab also has infinite order. Further- more, ab fixes 00 and has itself as a section at this vertex. Therefore G2369 is not contracting. 2371. Wreath recursion: a = σ(a, b), b = σ(c, a), c = (c, a). The element (c−1ab−1a)2 stabilizes the vertex 01 and its section at this vertex is equal to c−1ab−1a, which is nontrivial. Hence, c−1ab−1a has infinite order. 148 Classification of groups generated by automata Let t = b−1c−1a2c−1ba−1ca−1ca−2cbc−1ab−1a. Then t2 stabilizes the vertex 00 and t2 ∣ ∣ 00 = ta −1ba−1c . Hence, t has infinite order. Let s = b−1c−2a3 Since s8\|00100001 = t and s fixes the vertex 00100001 we see that s also has infinite order. Finally, s stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2371 is not contracting. 2372. Wreath recursion: a = σ(b, b), b = σ(c, a), c = (c, a). By Lemma 2 the elements b and ac act transitively on the levels of the tree and, hence, have infinite order. Since (c2) ∣ ∣ 100 = ac and the vertex 100 is fixed under the action of c2 we obtain that c also has infinite order. Finally, c stabilizes the vertex 0 and has itself as a section at this vertex. Therefore G2372 is not contracting. 2374 ∼= G821. Lamplighter group Z ≀ C2. Wreath recursion: a = σ(a, c), b = σ(c, a), c = (c, a). The states a and c form a 2-state automaton that generates the Lamp- lighter group (see Theorem 7). Since bc−1 = σ = c−1a, we have b = ac and G = 〈a, c〉. 2375. Wreath recursion: a = σ(b, c), b = σ(c, a), c = (c, a). The element (a−1c)2 stabilizes the vertex 01 and its section at this vertex is equal to a−1c. Hence, a−1c and c−1a have infinite order. Since c−1b−1ac−1a2 ∣ ∣ 00 = c−1a and the vertex 00 is fixed under the action of c−1b−1ac−1a2 we obtain that c−1b−1ac−1a2 also has infinite order. Fi- nally, c−1b−1ac−1a2 stabilizes the vertex 11 and has itself as a section at this vertex. Therefore G2375 is not contracting. 2376 ∼= G739 ∼= C2 ⋉ (C2 ≀ Z). Wreath recursion: a = σ(c, c), b = σ(c, a), c = (c, a). Since σ = bc−1, we have G2376 = 〈a, c, σ〉. We already proved that G972 = 〈a, c, σ〉. Therefore G2376 = G972 ∼= G739. 2388 ∼= G821. Lamplighter group Z ≀ C2. Wreath recursion: a = σ(c, a), b = σ(b, b), c = (c, a). The states a and c form a 2-state automaton generating the Lamp- lighter group (see Theorem 7) and b = σ = ac−1. 2391. Wreath recursion: a = σ(c, b), b = σ(b, b), c = (c, a). The element (c−1ba−1b)2 stabilizes the vertex 00 and its section at this vertex is equal to c−1ba−1b. Hence, c−1ba−1b has infinite order. Since (bc−2b)2 ∣ ∣ 000 = c−1ba−1b and 000 is fixed under the action of bc−2b we obtain that bc−2b also has infinite order. Finally, bc−2b stabilizes the vertex 1 and has itself as a section at this vertex. Therefore G2391 is not contracting. 2394 ∼= G820 ∼= D∞. Wreath recursion: a = σ(c, c), b = σ(b, b), c = (c, a). All generators have order 2, hence H = 〈ba, bc〉 is normal in G2394. Furthermore, ba = (bc, bc), bc = σ(bc, ba), and therefore H = G731 ∼= Z. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Thus G2394 = 〈b〉 ⋉H ∼= C2 ⋉ Z ∼= D∞ since (bc)b = (bc)−1. 2395. Wreath recursion: a = σ(a, a), b = σ(c, b), c = (c, a). By Lemma 2 the element ca acts transitively on the levels of the tree. The element (c−1a)2 stabilizes the vertex 0 and its section at this vertex is equal to c−1a. Hence, c−1a has infinite order. Since (b−1a) ∣ ∣ 0 = c−1a and 0 is fixed under the action of b−1a we obtain that b−1a also has infinite order. Finally, b−1a stabilizes the vertex 1 and has itself as a section at this vertex. Therefore G2395 is not contracting. Note that ab = (ac, ab), ac = σ(ac, 1) and ba = (ba, ca), ca = σ(1, ca), i.e., G2395 contains copies of G929. 2396. Boltenkov group. Wreath recursion: a = σ(b, a), b = σ(c, b), c = (c, a). This group was studied by A. Boltenkov (under direction of R. Grig- orchuk), who showed that the monoid generated by {a, b, c} is free, and the group G2396 is torsion free. Proposition 2. The monoid generated by a, b, and c is free. Proof. By way of contradiction, assume that there are some relations and let w = u be a relation for which max(\|w\|, \|u\|) minimal. We first consider the case when neither w nor u is empty. Because of cancelation laws, the words w and umust end in different letters. We have w = σw(w0, w1) = σu(u0, u1) = u, where σw, and σu are permutations in {1, σ}. Clearly, w0 = u0 and w1 = u1 must also be relations. Assume that w ends in b and u ends in c. Then w0 and u0 both end in c. Therefore, by minimality, w0 = u0 as words and \|u\| = \|w\|. Since b 6= c in G2396 the length of w and u is at least 2. We can recover the second to last letter in w and u. Indeed, the second to last letter in u0 can be only b or c (these are the possible sections at 0 of the three generators), while the second to last letter of w0 can be only a or b (these are the possible sections at 1 of the three generators). Therefore w0 = u0 = . . . bc, w = . . . bb , and u = . . . ac. Since bb 6= ac in G2396 (look at the action at level 1), the length of w and u must be at least 3. Continuing in the same fashion we obtain that w0 = u0 = b . . . bbc, w = . . . ababb, and u = . . . babac. Since the lengths of w and u are equal, they have different action on level 1, which is a contradiction. Assume that w ends in a and u ends in b or c. Then u0 and w0 end in b and c, respectively, and we may proceed as before. It remains to show that, say, u cannot be empty word. If this is the case then w0 = 1 = w1, implying that w0 = w1 is also a minimal relation. But this is impossible since both w0 and w1 are nonempty. For a group word w over {a, b, c}, define the exponent expa(w) of 150 Classification of groups generated by automata a in w as the sum of the exponents in all occurrences of a and a−1 in w. Define expb(w) and expc(w) in analogous way and let exp(w) = expa(w) + expb(w) + expc(w). Lemma 5. If w = 1 in G2396 then exp(w) = 0. Proof. By way of contradiction, assume otherwise and choose a freely reduced group word w over {a, b, c} such that w = 1 in G2396, exp(w) 6= 0, and w has minimal length among such words. If w = (w0, w1), w0 and w1 also represent 1 in G2396 and exp(w0) = exp(w1) = exp(w) 6= 0. Since the exponents is nonzero, the words w0 and w1 are nonempty and, by minimality, their length must be equal to \|w\|. Note that ac−1 = σ(bc−1, 1) and bc−1 = σ(1, ba−1). This implies that w cannot ac−1, bc−1, ca−1, or cb−1 as a subword (otherwise the length of w0 or w1 would be shorter than the length of w). By the same reason, w0 and w1 cannot have the above 4 words as subwords, which implies that w does not have ab−1 = (ab−1, bc−1) or its inverse ba−1 as a subword. Therefore w has the form w = W1(a −1, b−1, c−1)W2(a, b, c), and since w = 1 in G2396, we obtain a relation between positive words over {a, b, c}, which contradicts Proposition 2. Lemma 6. If w = 1 in G2396 then expa(w), expb(w) and expc(w) are even. Proof. Indeed, expa(w) + expb(w) must be even (since both a and b are active at the root). By Lemma 5, expc(w) must be even. If w = (w0, w1), then expa(w0)+expb(w0) and expa(w1)+expb(w1) must be even. Since expa(w) + expb(w) = expb(w0) + expb(w1), expa(w) + expc(w) = expa(w0)+expa(w1) we obtain that 2 expa(w)+expb(w)+expc(w) is even, which then implies that expb(w) is even. Finally, since both expb(w) and expc(w) are even, expa(w) must be even as well (by Lemma 5). Proposition 3. The group G2396 is torsion free. Proof. By way of contradiction, assume otherwise. Let w be an ele- ment of order 2. We may assume that w does not belong to the sta- bilizer of the first level (otherwise we may pass to a section of w). Let w = σ(w0, w1). Since w2 = (w1w0, w0w1) = 1, we have the modulo 2 equalities expb(w0w1) = expb(w0) + expb(w1) = expa(w) + expb(w). Since expb(w0w1) is even, expa(w) + expb(w) must be even, implying that w stabilizes level 1, a contradiction. Since b−1a = (c−1b, b−1a), the group G2396 is not contracting (our considerations above show that b−1a is not trivial and therefore has infi- nite order). I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, We have c−1bc−1a = (1, a−1bc−1b), ac−1bc−1 = (ba−1bc−1, 1), hence by Lemma 4 the group is not free. 2398. Dahmani group. Wreath recursion: a = σ(a, b), b = σ(c, b), c = (c, a). This group is self-replicating, not contracting, weakly regular branch group over its commutator subgroup. It was studied by Dahmani in [Dah05]. 2399. Wreath recursion: a = σ(b, b), b = σ(c, b), c = (c, a). By Lemma 2 the elements ca and c4bc2bc2b2cb2cb3acba2 act transi- tively on the levels of the tree and, hence, have infinite order. Since (cba)8 ∣ ∣ 000010001 = c4bc2bc2b2cb2cb3acba2 and vertex 000010001 is fixed un- der the action of (cba)8 we obtain that cba also has infinite order. Finally, cba stabilizes the vertex 01001 and has itself as a section at this vertex. Therefore G2399 is not contracting. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2401. Wreath recursion: a = σ(a, c), b = σ(c, b) and c = (c, a). The states a and c form a 2-state automaton generating the Lamp- lighter group (see Theorem 7). Hence, G2401 is neither torsion, nor con- tracting and has exponential growth. 2402. Wreath recursion: a = σ(b, c), b = σ(c, b), c = (c, a). The element (bc−1)2 stabilizes the vertex 00 and its section at this vertex is equal to bc−1. Hence, bc−1 has infinite order. We have c−2ba = (1, a−2b2), ac−2b = (ba−2b, 1), hence by Lemma 4 the group is not free. 2403 ∼= G2287. Wreath recursion: a = σ(c, c), b = σ(c, b), c = (c, a). The states a and c form a 2-state automaton generating D∞ (see Theorem 7). Also we have bc = σ(1, ba) and ba = (bc, 1). Therefore the elements bc and ba generate the Basilica group G852. By conjugating by g = (cg, g), we obtain a′ = σ, b′ = σ(1, c′b′), c′ = (c′, a′), where a′ = ag, b′ = bg, and c′ = cg. Therefore a′ = σ, b′ = σ(1, c′b′), c′b′ = σ(a′, b′), and G2402 is isomorphic to G2287, i.e., to IMG( z2+2 1−z2 ). 2422 ∼= G820 ∼= D∞. Wreath recursion: a = σ(a, a), b = σ(c, c), c = (c, a). The states a and c form a 2-state automaton generating D∞ (see Theorem 7) and b = aca. 152 Classification of groups generated by automata 2423. Wreath recursion: a = σ(b, a), b = σ(c, c), c = (c, a). Contains elements of infinite order by Lemma 1. In particular, ac has infinite order. Since c2 ∣ ∣ 100 = ac and the vertex 100 is fixed under the action of c2 we obtain that c also has infinite order. Since c = (c, a) the group is not contracting. We have c−1bc−1a = (1, a−1b), ac−1bc−1 = (ba−1, 1), hence by Lemma 4 the group is not free. 2424 ∼= G966. Wreath recursion a = σ(c, a), b = σ(c, c), c = (c, a). We have ac−1b = (c, c). Therefore G2424 = 〈a, ac−1b, c〉 = G966. 2426 ∼= G2277 ∼= C2 ⋉ (Z×Z). Wreath recursion: a = σ(b, b), b = σ(c, c), c = (c, a). Since all generators have order 2 the subgroup H = 〈ab, cb〉 is normal in G2426. Furthermore, ab = (bc, bc), cb = σ(ac, 1) = σ(ab(cb)−1, 1), so H is self-similar. Since acb = bca in G2426 we obtain ab · cb = abcaab = aacbab = cb · ab, hence, H is an abelian self-similar 2-generated group. Consider the 1 2 -endomorphism φ : StabH(1) → H, given by φ(g) = h, provided g = (h, ∗) and consider the linear map A : C 2 → C 2 induced by φ. It has the following matrix representation with respect to the basis corresponding to the generating set {ab, cb}: A = ( 0 1 2 −1 −1 2 ) . Its eigenvalues are not algebraic integers and, therefore, by [NS04], H is a free abelian group of rank 2. Finally, G2426 = 〈H, b〉 = 〈b〉 ⋉ H = C2 ⋉ (Z × Z), where b inverts the generators of H. This action coincides with the one for G2277, which proves that these groups are isomorphic. 2427. The element (bc−1)4 stabilizes the vertex 000 and its section at this vertex is equal to bc−1. Hence, bc−1 has infinite order. We have a−2bab−2ab = 1, and a and b do not commute, hence the group is not free. 2838 ∼= G848 ∼= C2 ≀ Z. Wreath recursion: a = σ(c, a), b = σ(a, a), c = (c, c). Since c is trivial, we have G = 〈a, ba−1〉, where a = σ(1, a) is the adding machine and ba−1 = (1, a). Therefore G2838 = G848. 2841. Wreath recursion: a = σ(c, b), b = σ(a, a), c = (c, c). The element c is trivial. Since a2 = (b, b), b2 = (a2, a2) and a2 is nontrivial, the elements a and b have infinite order. Also we have ba = (a, ab) and ab = (ba, a), hence ba has infinite order and G2841 is not contracting. We claim that the monoid generated by a and b is free. Hence, G2841 has exponential growth. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, Proof. We can first prove (analogous to G2851) that w 6= 1 for any nonempty word w ∈ {a, b}∗. By way of contradiction, let w and v be two nonempty words in {a, b}∗ with minimal \|w\| + \|v\| such that w = v in G2841. Assume that w ends with a and v ends with b. Consider the following cases. 1. If w = a then v\|0 = 1 in G2841 and v\|0 is nontrivial word. 2. If w ends with a2 then w\|1 = v\|1 in G2841, ∣ ∣w\|1 ∣ ∣+ ∣ ∣v\|1 ∣ ∣ < \|w\| + \|v\| and w\|1 ends with b, v\|1 with a. 3. If w ends with ba and v ends with ab, then w\|1 = v\|1 in G2841, ∣ ∣w\|1 ∣ ∣ + ∣ ∣v\|1 ∣ ∣ < \|w\| + \|v\| (because ∣ ∣v\|1 ∣ ∣ < \|v\|) and w\|1 ends with b, v\|1 with a. 4. If w ends with ba and v ends with b, then w\|1 = v\|1 in G2841, ∣ ∣w\|1 ∣ ∣+ ∣ ∣v\|1 ∣ ∣ ≤ \|w\|+\|v\| and w\|1 ends with ab, \|v1\| with a. Therefore, words v\|1 and w\|1 satisfy one of the first three cases. In all cases we obtain either a shorter relation, which contradicts to our assumption, or a relation of the form v = 1, which is also impossible. There are non-trivial group relations, e.g. a−1b−1a−2ba−1b−1aba2b−1ab = 1, while a and b do not commute, hence the group is not free. 2284 ∼= G730. Klein Group C2 × C2. Direct calculation. 2847 ∼= G929. Wreath recursion: a = σ(c, a), b = σ(b, a), c = (c, c). Since c is trivial, the generator a = σ(1, a) is the adding machine and b = σ(b, a). We have ab = (ab, a). Therefore G2847 = 〈a, ab〉 = G929. 2850. Wreath recursion: a = σ(c, b), b = σ(b, a), c = (c, c). Since c is trivial, we have a2 = (b, b), b2 = (ab, ba), ab = (b2, a) and ba = (a, b2). Therefore the elements a, b, ab and ba have infinite order. Since ba fixes the vertex 11 and has itself as a section at that vertex, G2850 is not contracting. The group is regular weakly branch over G′ 2850, since it is self- replicating and [b, a2] = (1, [a, b]). Semigroup 〈a, b〉 is free. Hence, G2850 has exponential growth. Proof. We can first prove (analogous G2851) that w 6= 1 for any nonempty word w ∈ {a, b}∗. By way of contradiction, let w and v be two nonempty words in {a, b}∗ with minimal \|w\| + \|v\| such that w = v in G2850. Assume that w ends with a and v ends with b. Consider the following cases. 154 Classification of groups generated by automata 1. If w = a then v\|0 = 1 in G and v\|0 is nontrivial word. 2. If w ends with a2 then w\|1 = v\|1 in G, ∣ ∣w\|1 ∣ ∣+ ∣ ∣v\|1 ∣ ∣ < \|w\|+ \|v\| and w\|1 ends with b, v\|1 with a. 3. If w ends with ba then w\|0 = v\|0 in G, ∣ ∣w\|0 ∣ ∣+ ∣ ∣v\|0 ∣ ∣ < \|w\|+ \|v\| and w\|0 ends with a, v\|0 with b. In all cases we obtain either a shorter relation, which contradicts to our assumption, or a relation of the form v = 1, which is also impossible. Since a−4bab−1a2b−1ab = 1 and a and b do not commute, the group is not free. 2851 ∼= G929. Wreath recursion: a = σ(a, c), b = σ(b, a), c = (c, c). The automorphism c is trivial. Therefore a = σ(a, 1) is the inverse of the adding machine. Since ba−1 = (a, ba−1), the order of ba−1 is infinite and G2851 is not contracting. Since G2851 is self-replicating and [a2, b] = ([a, b], 1), the group is a regular weakly branch group over its commutator. The monoid 〈a, b〉 is free. Proof. By way of contradiction, assume that w be a nonempty word over {a, b} such that w = 1 in G2851 and w has the smallest length among all such words. The word w must contain both a and b (since they have infinite order). Therefore, one of the projections of w is be shorter than w, nonempty, and represents the identity in G2851, a contradiction. Assume now that w and v are two nonempty words over {a, b} such that w = v in G2851 and they are chosen so that the sum \|w\| + \|v\| is minimal. Assume that w ends in a and v ends in b. Then - if w ends in a2, then w0 is a nonempty word that is shorter than w ending in a, while v0 is a nonempty word of length no greater than \|v\| ending in b. Since w0 = v0 in G2851, this contradicts the minimality assumption. - if w ends in ba, then w1 is a word that is shorter than w ending in b, while v1 is a nonempty word of length no greater than \|v\| ending in a. Since w1 = v1 in G2851, this contradicts the minimality assumption. - if w = a then v1 = 1 in G and v1 is a nonempty word. Thus we obtain a relation v1 = 1 in G2851, a contradiction. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, This shows that G has exponential growth, while the orbital Schreier graph Γ(G, 000 . . .) has intermediate growth (see [BH05, BCSN]). The groupsG2851 andG929 coincide as subgroups of Aut(X∗). Indeed, a−1 = σ(1, a−1) is the adding machine and b−1a = (b−1a, a−1), showing that G929 = 〈a−1, b−1a〉 = G2851. 2852 ∼= G849. Wreath recursion: a = σ(b, c), b = σ(b, a), c = (c, c). The automorphism c is trivial. Therefore a = σ(b, 1), a2 = (b, b) and ab = (b, ba), which implies that G2852 is self-replicating and level transitive. The group G2852 is a regular weakly branch group over its commuta- tor. This follows from [a−1, b] · [b, a] = ([a, b], 1), together with the self- replicating property and the level transitivity. Moreover, the commutator is not trivial, since G2852 is not abelian (note that [b, a] = (b−1ab, a−1) 6= 1). We have b2 = (ab, ba), ba = (ab, b), and ab = (b, ba). Therefore b2 fixes the vertex 00 and has b as a section at this vertex. Therefore b has infinite order (since it is nontrivial), and so do ab and a (since a2 = (b, b)). Since ab fixes the vertex 10 and has itself as a section at that vertex, G2852 is not contracting. The monoid generated by a and b is free (and therefore the group has exponential growth). Proof. By way of contradiction assume that w is a word of minimal length over all nonempty words over {a, b} such that w = 1 in G2851. Then w must have occurrences of both a and b (since both have infinite order). This implies that one of the sections of w is shorter than w (since a\|1 is trivial), nonempty (since both b\|0 and b\|1 are nontrivial), and represents the identity in G2851, a contradiction. Assume now that there are two nonempty words w, v ∈ {a, b}∗ such that w = u in G2852 and choose such words with minimal sum \|w\| + \|v\|. Let w = σw(w0, w1) and u = σu(u0, u1), where σw, σw ∈ {1, σ}. Assume that w ends in a and v ends in b (they must end in different letters because of the cancelation property and the minimality of the choice). Then w1 = v1 in G2851, the word v1 is nonempty, \|v1\| ≤ \|v\|, and \|w1\| < \|w\|. Thus we either obtain a contradiction with the minimality of the choice of w and v or we obtain a relation of the type v1 = 1, also a contradiction. See G849 for an isomorphism between G2852 and G849. If we conjugate the generators of G2852 by the automorphism µ = σ(bµ, µ), we obtain the wrath recursion x = σ(y, 1), y = σ(xy, 1), 156 Classification of groups generated by automata where x = aµ and y = bµ. Further, y = σ(xy, 1), xy = (xy, y), and the last recursion defines the automaton 933. Therefore G2852 ∼= G933. 2853 ∼= IMG ( ( z−1 z+1 )2 ) . Wreath recursion a = σ(c, c), b = σ(b, a) and c = (c, c). The automorphism c is trivial and a = σ. It is shown in [BN06] that IMG ( ( z−1 z+1 )2 ) is generated by α = σ(1, β) and β = (α−1β−1, α). We have then βα = σ(α, α−1). If we conjugate by γ = (γ, αγ), we obtain the wreath recursion A = σ, B = σ(B−1, A) where A = (βα)γ and B = αγ . The group 〈A,B〉 is conjugate to G2853 by the element δ = (δ1, δ1), where δ1 = σ(δ, δ) (this is the automorphism of the tree changing the letters on even positions). Therefore G2852 ∼= IMG ( ( z−1 z+1 )2 ) and the limit space of G2852 is the Julia set of the rational map z 7→ ( z−1 z+1 )2 . Note that G2853 is contained in G775 as a subgroup of index 2. There- fore it is virtually torsion free (it contains the torsion free subgroup H mentioned in the discussion of G775 as a subgroup of index 2) and is a weakly branch group over H ′′. The diameters of the Schreier graphs on the levels grow as √ 2 n and have polynomial growth of degree 2 (see [BN, Bon07]). 2854 ∼= G847 ∼= D4. Wreath recursion: a = σ(a, a), b = σ(c, a), c = (c, c). Direct calculation. 2860 ∼= G2212. Klein bottle group 〈s, t ∣ ∣ s2 = t2〉. Wreath recursion: a = σ(a, c), b = σ(c, a), c = (c, c)〉. Note that c is trivial and therefore a = σ(a, 1) and b = σ(1, a). The element a has infinite order since a is inverse of the adding machine. Let us prove that G2860 ∼= H = 〈s, t ∣ ∣ s2 = t2〉. Indeed, the relation a2 = b2 is satisfied, so G2860 is a homomorphic image of H with respect to the homomorphism induced by s 7→ a and t 7→ b. Each element of H can be written in the form tr(st)lsn, n ∈ Z, l ≥ 0, r ∈ {0, 1}. It suffices to prove that images of these words (except for the identity word, of course) represent nonidentity elements in G2860. We have a2n = (an, an), a2n+1 = σ(an+1, an), (ab)l = (1, a2l). We only need to check words of even length (those of odd length act I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, nontrivially on level 1). We have (ab)ℓa2n = (an, an+2ℓ) 6= 1 in G if n 6= 0 or ℓ 6= 0, since a has infinite order. On the other hand, b(ab)la2n+1 = (an+1+2l+1, an) = 1 if and only if n = 0 and l = −1, which is not the case, because l must be nonnegative. This finishes the proof. 2861 ∼= G731 ∼= Z. Wreath recursion: a = σ(b, c), b = σ(c, a), c = (c, c)〉. Since c is trivial, ba = (ab, 1), ab = (1, ba), which yields a = b−1. Also a2n = (bn, bn), b2n = (an, an) and a2n+1 6= 1, b2n+1 6= 1. Thus a has infinite order and G2861 ∼= Z. 2862 ∼= G847 ∼= D4. Wreath recursion: a = σ(c, c), b = σ(c, a), c = (c, c)〉. Direct calculation. 2874 ∼= G820 ∼= D∞. Wreath recursion: a = σ(a, c), b = σ(b, b), c = (c, c). Since c is trivial, G2874 = 〈b, ba〉. Since ba = (ba, b), the elements b and ba form a 2-state automaton generating D∞ (see Theorem 7). 2880 ∼= G730. Klein Group C2 × C2. Wreath recursion: a = σ(c, c), b = σ(b, b), c = (c, c). Direct calculation. 2887 ∼= G731 ∼= Z. Wreath recursion: a = σ(a, c), b = σ(c, b), c = (c, c). Note that c is trivial, b is the adding machine and a = b−1. 2889 ∼= G848 ∼= C2 ≀ Z. Wreath recursion: a = σ(c, c), b = σ(c, b), c = (c, c). Note that c is trivial. Since b is the adding machine and ab = (1, b), we have G2889 = 〈b, ab〉 = G848. References [Adi79] S. I. Adian. The Burnside problem and identities in groups, volume 95 of Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas]. Springer-Verlag, Berlin, 1979. [Ale72] S. V. Alešin. Finite automata and the Burnside problem for periodic groups. Mat. Zametki, 11:319–328, 1972. [Ale83] S. V. Aleshin. A free group of finite automata. Vestnik Moskov. Univ. Ser. I Mat. Mekh., (4):12–14, 1983. [Bar03] Laurent Bartholdi. A Wilson group of non-uniformly exponential growth. C. R. Math. Acad. Sci. Paris, 336(7):549–554, 2003. [BCSN] Ievgen Bondarenko, Tullio Checcherini-Silberstein, and Volodymyr Nekra- shevych. Amenable graphs with dense holonomy and no compact isometry groups. In preparation. [BG00a] L. Bartholdi and R. I. Grigorchuk. On the spectrum of Hecke type operators related to some fractal groups. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):5–45, 2000. 158 Classification of groups generated by automata [BG00b] Laurent Bartholdi and Rostislav I. Grigorchuk. Lie methods in growth of groups and groups of finite width. In Michael Atkinson et al., editor, Computational and Geometric Aspects of Modern Algebra, volume 275 of London Math. Soc. Lect. Note Ser., pages 1–27. Cambridge Univ. Press, Cambridge, 2000. [BGK+a] I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekra- shevych, D. Savchuk, and Z. Šunić. Groups generated by 3-state automata over 2-letter alphabet, I. accepted in Sao Paolo Journal of Mathematical Sciences, http://xxx.arxiv.org/abs/0704.3876. [BGK+b] I. Bondarenko, R. Grigorchuk, R. Kravchenko, Y. Muntyan, V. Nekra- shevych, D. Savchuk, and Z. Šunić. Groups generated by 3-state automata over 2-letter alphabet, II. accepted in Journal of Mathematical Sciences (N.Y.), http://xxx.arxiv.org/abs/math/0612178. [BGŠ03] Laurent Bartholdi, Rostislav I. Grigorchuk, and Zoran Šuniḱ. Branch groups. In Handbook of algebra, Vol. 3, pages 989–1112. North-Holland, Amsterdam, 2003. [BH05] Itai Benjamini and Christopher Hoffman. ω-periodic graphs. Electron. J. Combin., 12:Research Paper 46, 12 pp. (electronic), 2005. [BKN] Laurent Bartholdi, Vadim Kaimanovich, and Volodymyr Nekrashevych. On amenability of automata groups. arXiv:0802.2837. [BN] Ievgen Bondarenko and Volodymyr Nekrashevych. Growth of Schreier graphs of groups generated by bounded automata. in preparation. [BN06] Laurent I. Bartholdi and Volodymyr V. Nekrashevych. Thurston equiva- lence of topological polynomials. Acta Math., 197(1):1–51, 2006. [BN07] Laurent Bartholdi and Volodymyr Nekrashevych. Iterated monodromy groups of quadratic polynomials, I, 2007. acceted in Groups, Geometry, and Dynamics. [Bon07] Ievgen Bondarenko. Groups generated by bounded automata and their Schreier graphs. PhD dissertation, Texas A&M University, 2007. [BRS06] L. Bartholdi, I. I. Reznykov, and V. I. Sushchansky. The smallest Mealy automaton of intermediate growth. J. Algebra, 295(2):387–414, 2006. [BŠ06] Laurent I. Bartholdi and Zoran Šuniḱ. Some solvable automaton groups. In Topological and Asymptotic Aspects of Group Theory, volume 394 of Contemp. Math., pages 11–29. Amer. Math. Soc., Providence, RI, 2006. [BSV99] A. M. Brunner, Said Sidki, and Ana Cristina Vieira. A just nonsolvable torsion-free group defined on the binary tree. J. Algebra, 211(1):99–114, 1999. [BV05] Laurent Bartholdi and Bálint Virág. Amenability via random walks. Duke Math. J., 130(1):39–56, 2005. [CM82] Bruce Chandler and Wilhelm Magnus. The history of combinatorial group theory, volume 9 of Studies in the History of Mathematics and Physical Sciences. Springer-Verlag, New York, 1982. [Dah05] François Dahmani. An example of non-contracting weakly branch automa- ton group. In Geometric methods in group theory, volume 372 of Contemp. Math., pages 219–224. Amer. Math. Soc., Providence, RI, 2005. [Day57] Mahlon M. Day. Amenable semigroups. Illinois J. Math., 1:509–544, 1957. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, [Eil76] Samuel Eilenberg. Automata, languages, and machines. Vol. B. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976. [EP84] M. Edjvet and Stephen J. Pride. The concept of “largeness” in group theory. II. In Groups—Korea 1983 (Kyoungju, 1983), volume 1098 of Lecture Notes in Math., pages 29–54. Springer, Berlin, 1984. [GL02] Rostislav Grigorchuk and Igor Lysionok. Burnside problem. In Alexan- der V. Mikhalev and Günter F. Pilz, editors, The concise handbook of al- gebra, pages 111–115. Kluwer Academic Publishers, Dordrecht, 2002. [GLSŻ00] Rostislav I. Grigorchuk, Peter Linnell, Thomas Schick, and Andrzej Żuk. On a question of Atiyah. C. R. Acad. Sci. Paris Sér. I Math., 331(9):663– 668, 2000. [Glu61] V. M. Glushkov. Abstract theory of automata. Uspekhi mat. nauk., 16(5):3– 62, 1961. (in Russian). [GM05] Yair Glasner and Shahar Mozes. Automata and square com- plexes. Geom. Dedicata, 111:43–64, 2005. (available at http://arxiv.org/abs/math.GR/0306259 ). [GN07] Rostislav Grigorhuk and Volodymyr Nekrashevych. Self-similar groups, operator algebras and schur complement. J. Modern Dyn., 1(3):323–370, 2007. [GNS00] R. I. Grigorchuk, V. V. Nekrashevich, and V. I. Sushchanskĭı. Automata, dynamical systems, and groups. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):134–214, 2000. [GNŠ06a] Rostislav Grigorchuk, Volodymyr Nekrashevych, and Zoran Šunić. Hanoi towers group on 3 pegs and its pro-finite closure. Oberwolfach Reports, 25:15–17, 2006. [GNŠ06b] Rostislav Grigorchuk, Volodymyr Nekrashevych, and Zoran Šunić. Hanoi towers groups. Oberwolfach Reports, 19:11–14, 2006. [Gol68] E. S. Golod. Some problems of Burnside type. In Proc. Internat. Congr. Math. (Moscow, 1966), pages 284–289. Izdat. “Mir”, Moscow, 1968. [GP72] F. Gecseg and I. Peák. Algebraic theory of automata. Akadémiai Kiadó, Budapest, 1972. Disquisitiones Mathematicae Hungaricae, 2. [Gri80] R. I. Grigorčuk. On Burnside’s problem on periodic groups. Funktsional. Anal. i Prilozhen., 14(1):53–54, 1980. [Gri83] R. I. Grigorchuk. On the Milnor problem of group growth. Dokl. Akad. Nauk SSSR, 271(1):30–33, 1983. [Gri84] R. I. Grigorchuk. Degrees of growth of finitely generated groups and the theory of invariant means. Izv. Akad. Nauk SSSR Ser. Mat., 48(5):939–985, 1984. [Gri85] R. I. Grigorchuk. Degrees of growth of p-groups and torsion-free groups. Mat. Sb. (N.S.), 126(168)(2):194–214, 286, 1985. [Gri89] R. I. Grigorchuk. On the Hilbert-Poincaré series of graded algebras that are associated with groups. Mat. Sb., 180(2):207–225, 304, 1989. [Gri98] R. I. Grigorchuk. An example of a finitely presented amenable group that does not belong to the class EG. Mat. Sb., 189(1):79–100, 1998. 160 Classification of groups generated by automata [Gri99] R. I. Grigorchuk. On the system of defining relations and the Schur mul- tiplier of periodic groups generated by finite automata. In Groups St. An- drews 1997 in Bath, I, volume 260 of London Math. Soc. Lecture Note Ser., pages 290–317. Cambridge Univ. Press, Cambridge, 1999. [Gri00] R. I. Grigorchuk. Just infinite branch groups. In New horizons in pro-p groups, volume 184 of Progr. Math., pages 121–179. Birkhäuser Boston, Boston, MA, 2000. [GS83a] N. Gupta and Said Sidki. Some infinite p-groups. Algebra i Logika, 22(5):584–589, 1983. [GS83b] Narain Gupta and Säıd Sidki. On the Burnside problem for periodic groups. Math. Z., 182(3):385–388, 1983. [GŠ06] Rostislav Grigorchuk and Zoran Šuniḱ. Asymptotic aspects of Schreier graphs and Hanoi Towers groups. C. R. Math. Acad. Sci. Paris, 342(8):545– 550, 2006. [GŠ07] Rostislav Grigorchuk and Zoran Šunić. Self-similarity and branching in group theory. In Groups St. Andrews 2005, I, volume 339 of London Math. Soc. Lecture Note Ser., pages 36–95. Cambridge Univ. Press, Cambridge, 2007. [GSŠ07] Rostislav Grigorchuk, Dmytro Savchuk, and Zoran Šunić. The spectral problem, substitutions and iterated monodromy. In Probability and math- ematical physics, volume 42 of CRM Proc. Lecture Notes, pages 225–248. Amer. Math. Soc., Providence, RI, 2007. [Gup89] Narain Gupta. On groups in which every element has finite order. Amer. Math. Monthly, 96(4):297–308, 1989. [GW00] R. I. Grigorchuk and J. S. Wilson. The conjugacy problem for certain branch groups. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):215–230, 2000. [GW03] R. I. Grigorchuk and J. S. Wilson. A structural property concerning ab- stract commensurability of subgroups. J. London Math. Soc. (2), 68(3):671– 682, 2003. [GŻ99] Rostislav I. Grigorchuk and Andrzej Żuk. On the asymptotic spectrum of random walks on infinite families of graphs. In Random walks and discrete potential theory (Cortona, 1997), Sympos. Math., XXXIX, pages 188–204. Cambridge Univ. Press, Cambridge, 1999. [GŻ01] Rostislav I. Grigorchuk and Andrzej Żuk. The lamplighter group as a group generated by a 2-state automaton, and its spectrum. Geom. Dedicata, 87(1- 3):209–244, 2001. [GŻ02a] Rostislav I. Grigorchuk and Andrzej Żuk. On a torsion-free weakly branch group defined by a three state automaton. Internat. J. Algebra Comput., 12(1-2):223–246, 2002. [GŻ02b] Rostislav I. Grigorchuk and Andrzej Żuk. Spectral properties of a torsion- free weakly branch group defined by a three state automaton. In Compu- tational and statistical group theory (Las Vegas, NV/Hoboken, NJ, 2001), volume 298 of Contemp. Math., pages 57–82. Amer. Math. Soc., Providence, RI, 2002. [Hoř63] Jǐŕı Hořeǰs. Transformations defined by finite automata. Problemy Kiber- net., 9:23–26, 1963. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, [KAP85] V. B. Kudryavtsev, S. V. Aleshin, and A. S. Podkolzin. Vvedenie v teoriyu avtomatov. “Nauka”, Moscow, 1985. [KM82] M. I. Kargapolov and Yu. I. Merzlyakov. Osnovy teorii grupp. “Nauka”, Moscow, third edition, 1982. [Kos90] A. I. Kostrikin. Around Burnside, volume 20 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag, Berlin, 1990. Translated from the Russian and with a preface by James Wiegold. [KSS06] Mark Kambites, Pedro V. Silva, and Benjamin Steinberg. The spectra of lamplighter groups and Cayley machines. Geom. Dedicata, 120:193–227, 2006. [Leo98] Yu. G. Leonov. The conjugacy problem in a class of 2-groups. Mat. Zametki, 64(4):573–583, 1998. [LN02] Yaroslav Lavreniuk and Volodymyr Nekrashevych. Rigidity of branch groups acting on rooted trees. Geom. Dedicata, 89:159–179, 2002. [Lys85] I. G. Lysënok. A set of defining relations for the Grigorchuk group. Mat. Zametki, 38(4):503–516, 634, 1985. [Mer83] Yu. I. Merzlyakov. Infinite finitely generated periodic groups. Dokl. Akad. Nauk SSSR, 268(4):803–805, 1983. [Mil68] J. Milnor. Problem 5603. Amer. Math. Monthly, 75:685–686, 1968. [MNS00] O. Macedońska, V. Nekrashevych, and V. Sushchansky. Commensurators of groups and reversible automata. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (12):36–39, 2000. [MS08] Y. Muntyan and D. Savchuk. AutomGrp – GAP package for computations in self-similar groups and semigroups, Version 1.1.2, 2008. (available at http://finautom.sourceforge.net). [Nek05] Volodymyr Nekrashevych. Self-similar groups, volume 117 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005. [Nek07a] V. Nekrashevych. Free subgroups in groups acting on rooted trees, 2007. preprint. [Nek07b] Volodymyr Nekrashevych. A group of non-uniform exponential growth lo- cally isomorphic to IMG(z2 + i), 2007. preprint. [Neu86] Peter M. Neumann. Some questions of Edjvet and Pride about infinite groups. Illinois J. Math., 30(2):301–316, 1986. [NS04] V. Nekrashevych and S. Sidki. Automorphisms of the binary tree: state- closed subgroups and dynamics of 1/2-endomorphisms. volume 311 of Lon- don Math. Soc. Lect. Note Ser., pages 375–404. Cambridge Univ. Press, 2004. [NT08] Volodymyr Nekrashevych and Alexander Teplyaev. Groups and analysis on fractals. to appear in Proceedings of “Analysis on Graphs and Applications”, 2008. [Oli98] Ricardo Oliva. On the combinatorics of extenal rays in the dynamics of the complex Henon map. PhD thesis, Cornell University, 1998. 162 Classification of groups generated by automata [Per00] E. L. Pervova. Everywhere dense subgroups of a group of tree auto- morphisms. Tr. Mat. Inst. Steklova, 231(Din. Sist., Avtom. i Beskon. Gruppy):356–367, 2000. [Per02] E. L. Pervova. The congruence property of AT-groups. Algebra Logika, 41(5):553–567, 634, 2002. [Pri80] Stephen J. Pride. The concept of “largeness” in group theory. In Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), vol- ume 95 of Stud. Logic Foundations Math., pages 299–335. North-Holland, Amsterdam, 1980. [Roz93] A. V. Rozhkov. Centralizers of elements in a group of tree automorphisms. Izv. Ross. Akad. Nauk Ser. Mat., 57(6):82–105, 1993. [Roz98] A. V. Rozhkov. The conjugacy problem in an automorphism group of an infinite tree. Mat. Zametki, 64(4):592–597, 1998. [RS] John Rhodes and Pedro V. Silva. An algebraic analysis of turing ma- chines and cook’s theorem leading to a profinite fractal differential equation. preprint. [RS02a] I. I. Reznikov and V. I. Sushchanskĭı. Growth functions of two-state au- tomata over a two-element alphabet. Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki, (2):76–81, 2002. [RS02b] I. I. Reznikov and V. I. Sushchanskĭı. Two-state Mealy automata of inter- mediate growth over a two-letter alphabet. Mat. Zametki, 72(1):102–117, 2002. [RS02c] I. I. Reznykov and V. I. Sushchansky. 2-generated semigroup of automatic transformations whose growth is defined by Fibonachi series. Mat. Stud., 17(1):81–92, 2002. [Sav03] Dmytro M. Savchuk. On word problem in contracting automorphism groups of rooted trees. Vı̄sn. Kïıv. Un̄ıv. Ser. F̄ız.-Mat. Nauki, (1):51–56, 2003. [Sid87a] Said Sidki. On a 2-generated infinite 3-group: subgroups and automor- phisms. J. Algebra, 110(1):24–55, 1987. [Sid87b] Said Sidki. On a 2-generated infinite 3-group: the presentation problem. J. Algebra, 110(1):13–23, 1987. [Sid00] Said Sidki. Automorphisms of one-rooted trees: growth, circuit structure, and acyclicity. J. Math. Sci. (New York), 100(1):1925–1943, 2000. Algebra, 12. [Sid04] Said Sidki. Finite automata of polynomial growth do not generate a free group. Geom. Dedicata, 108:193–204, 2004. [Sus79] V. I. Sushchansky. Periodic permutation p-groups and the unrestricted Burnside problem. DAN SSSR., 247(3):557–562, 1979. (in Russian). [VV05] M. Vorobets and Ya. Vorobets. On a free group of transformations de- fined by an automaton, 2005. To appear in Geom. Dedicata. (available at http://arxiv.org/abs/math/0601231 ). [Wil04a] John S. Wilson. Further groups that do not have uniformly exponential growth. J. Algebra, 279(1):292–301, 2004. [Wil04b] John S. Wilson. On exponential growth and uniformly exponential growth for groups. Invent. Math., 155(2):287–303, 2004. I. Bondarenko, R. Grigorchuk, R. Kravchenko, Ye. Muntyan, V. Nekrashevych, [Wol02] Stephen Wolfram. A new kind of science. Wolfram Media, Inc., Champaign, IL, 2002. [WZ97] J. S. Wilson and P. A. Zalesskii. Conjugacy separability of certain torsion groups. Arch. Math. (Basel), 68(6):441–449, 1997. [Zar64] V. P. Zarovny̆ı. On the group of automatic one-to-one mappings. Dokl. Akad. Nauk SSSR, 156:1266–1269, 1964. [Zar65] V. P. Zarovny̆ı. Automata substitutions and wreath products of groups. Dokl. Akad. Nauk SSSR, 160:562–565, 1965. [Zel91] Efim I. Zelmanov. On the restricted Burnside problem. In Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), pages 395–402, Tokyo, 1991. Math. Soc. Japan. Contact information I. Bondarenko Kyiv Taras Shevchenko University, Ukraine E-Mail: ievgbond@gmail.com R. Grigorchuk Texas A&M University, USA E-Mail: grigorch@math.tamu.edu URL: www.math.tamu.edu/˜grigorch R. Kravchenko Texas A&M University, USA E-Mail: rkchenko@math.tamu.edu Y. Muntyan Texas A&M University, USA E-Mail: muntyan@math.tamu.edu URL: www.math.tamu.edu/˜muntyan V. Nekrashevych Texas A&M University, USA E-Mail: nekrash@math.tamu.edu URL: www.math.tamu.edu/˜nekrash D. Savchuk Texas A&M University, USA E-Mail: savchuk@math.tamu.edu URL: www.math.tamu.edu/˜savchuk Z. Šunić Texas A&M University, USA E-Mail: sunic@math.tamu.edu URL: www.math.tamu.edu/˜sunik Received by the editors: 18.12.2007 and in final form 15.02.2007.

On classification of groups generated by 3-state automata over a 2-letter alphabet

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