On simple groups of large exponents
It is shown that the set of pairwise non-isomorphic 2-generated simple groups of exponent n (n ≥ 2⁴⁸ and n is odd or divisible by 2⁹ ) is of cardinality continuum. As a corollary, for any sufficiently large n the set of pairwise non-isomorphic 2-generated groups of exponent n is of cardina...
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irk-123456789-1566012019-06-19T01:26:21Z On simple groups of large exponents Sonkin, D. It is shown that the set of pairwise non-isomorphic 2-generated simple groups of exponent n (n ≥ 2⁴⁸ and n is odd or divisible by 2⁹ ) is of cardinality continuum. As a corollary, for any sufficiently large n the set of pairwise non-isomorphic 2-generated groups of exponent n is of cardinality continuum. 2004 Article On simple groups of large exponents / D. Sonkin // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 79–105. — Бібліогр.: 10 назв. — англ. 1726-3255 2000 Mathematics Subject Classification: 20F50; 20F05, 20F06. http://dspace.nbuv.gov.ua/handle/123456789/156601 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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It is shown that the set of pairwise non-isomorphic
2-generated simple groups of exponent n (n ≥ 2⁴⁸ and n is odd or
divisible by 2⁹
) is of cardinality continuum. As a corollary, for any
sufficiently large n the set of pairwise non-isomorphic 2-generated
groups of exponent n is of cardinality continuum. |
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Sonkin, D. |
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Sonkin, D. On simple groups of large exponents Algebra and Discrete Mathematics |
| author_facet |
Sonkin, D. |
| author_sort |
Sonkin, D. |
| title |
On simple groups of large exponents |
| title_short |
On simple groups of large exponents |
| title_full |
On simple groups of large exponents |
| title_fullStr |
On simple groups of large exponents |
| title_full_unstemmed |
On simple groups of large exponents |
| title_sort |
on simple groups of large exponents |
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Інститут прикладної математики і механіки НАН України |
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2004 |
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http://dspace.nbuv.gov.ua/handle/123456789/156601 |
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On simple groups of large exponents / D. Sonkin // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 79–105. — Бібліогр.: 10 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT sonkind onsimplegroupsoflargeexponents |
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2025-07-14T08:59:45Z |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 4. (2004). pp. 79 – 105
c© Journal “Algebra and Discrete Mathematics”
On simple groups of large exponents
Dmitriy Sonkin
Communicated by A. Yu. Olshanskii
Abstract. It is shown that the set of pairwise non-isomorphic
2-generated simple groups of exponent n (n ≥ 248 and n is odd or
divisible by 29) is of cardinality continuum. As a corollary, for any
sufficiently large n the set of pairwise non-isomorphic 2-generated
groups of exponent n is of cardinality continuum.
Introduction
B.H. Neumann [9] showed that the set of all pairwise non-isomorphic 2-
generated groups is of cardinality continuum. It is known that so is the
set of pairwise non-isomorphic 2-generated simple groups. Moreover, for
any sufficiently large prime number p the set of pairwise non-isomorphic
2-generated simple groups satisfying the identity xp = 1 is of cardinality
continuum (for a detailed discussion see [7] and [2]).
In this paper, we prove that for almost all values of n the same is
true about the set of pairwise non-isomorphic 2-generated groups of ex-
ponent n.
The present investigation is heavily dependent on the technique ex-
posed in S. V. Ivanov’s paper [3], the main result of which is that the
free m-generated Burnside group B(m, n) of even exponent n is infinite
provided n ≥ 248, n is either odd or divisible by 29 and m > 1 (see also
[5], and [1], [6], [7] for the earlier results dealing with odd exponents).
This work was supported in part by the NSF grant 0072307 of A.Yu. Ol’shanskii
and M.V. Sapir.
2000 Mathematics Subject Classification: 20F50; 20F05, 20F06.
Key words and phrases: Burnside group, van Kampen diagram, graded dia-
gram, graded presentation, non-isomorphic 2-generated torsion simple groups.
80 On simple groups of large exponents
To achieve our goal we introduce a family of torsion groups {GT }.
We shall start with the absolutely free group F of rank 2. First we shall
impose on F some infinite set of relations, left-hand-sides of which are
taken from a certain infinite set T of words in a 2-letter alphabet satis-
fying strong small cancellation and aperiodicity conditions (see Lemma
0.1). Then, one-by-one we will impose the relations that will guarantee
that the presentation we have obtained is a presentation of a member
of the Burnside variety of exponent n, where n ≥ 248 and n is odd or
divisible by 29. Constructing the presentations of groups GT in Section
2 below we constantly refer the reader to [3].
In Theorem A below we collect some properties of groups GT . Note,
in particular, that finite subgroups of GT behave in the same way as those
of a free Burnside group of the corresponding exponent (see [3], Theorem
A(c)).
Theorem A. For every set T (finite or infinite) satisfying conditions
(a), (b) and (c) of Lemma 0.1 the group GT has the following properties:
(1) GT is a 2-generated infinite group belonging to the Burnside variety
of exponent n.
(2) Let n = n1n2, where n1 is the maximal odd divisor of n. Then
every finite subgroup of GT is isomorphic to a subgroup of D(2n1)×
D(2n2)
k, for some k, where D(2m) is a dihedral group of order 2m.
(3) The center of GT is trivial.
Furthermore, the set of pairwise non-isomorphic groups among {GT ′},
T ′ ⊆ T , is of cardinality continuum provided T is infinite. If T ′ is a
recursive subset of T , then the word and conjugacy problems are solvable
in GT ′.
In Section 3 we pick special sets Tα and show that the set of pair-
wise non-isomorphic simple quotients of the groups GTα
is of cardinality
continuum.
Theorem B. For any sufficiently large exponent n the set of pairwise
non-isomorphic 2-generated simple groups satisfying the identity xn = 1
is of cardinality continuum.
In the proof of this theorem given in Section 3 we assume that n ≥ 248
and n is either odd or divisible by 29. The statement for any multiple of
such n clearly follows.
D. Sonkin 81
1. T -relators
We assume that the reader is familiar with the terminology from [7]. For
the numerical values of auxiliary parameters (α, β, etc.) we refer to [3].
Certain sets of aperiodic words satisfying small cancellation conditions
were constructed in [8] and [10]. We refer to [10] for the proof of the
following
Lemma 0.1. For given n and ξ > 0 there exists an infinite set of
positive words T = {B1, B2, . . . } in the alphabet {a1, a2}, satisfying the
following properties:
(a) Suppose a cyclic shift of some word B±1
i contains a B-periodic sub-
word U of length greater than (1 + ξ)|B|. Then |B| < ξ−3/2 and
|U | < 11|B| < 11ξ−3/2.
(b) The symmetrized set obtained from the set T satisfies the small
cancellation condition C ′( ξ
10).
(c) |Bi| ≥ n2, i = 1, 2, . . . .
According to the choice of the auxiliary parameters (βn2 > 11ξ−3/2,
β > ξ) the next lemma immediately follows from Lemma 0.1.
Lemma 0.2. Let U be a B-periodic subword of a cyclic shift of some
word B±1
i of length |U | > β|Bi|. Then |U | < (1 + ξ)|B| and |Bi| <
(1 + ξ)β−1|B|.
Define G(0) = F (a1, a2) to be absolutely free group and set
G(1/2) = 〈a1, a2 | B = 1, B ∈ T 〉 . (1)
Arguing as for C ′(1/8)-groups (see [4], Chapter V, Theorem 10.1), in
view of Lemma 0.1 we obtain the following
Lemma 0.3. The group G(1/2) is torsion free.
2. Inductive construction of group GT
For every i, i = 0, 1/2, 1, 2, . . . , we shall define the group G(i) of rank
i. The groups G(0) and G(1/2) are already defined. Following [3], the
elements of F (a1, a2) and of its quotients are referred to as words (in
the alphabet {a±1
1 , a±1
2 }). Let ’≺’ be a total order on the set of words
82 On simple groups of large exponents
over {a±1
1 , a±1
2 }, such that |X| < |Y | implies |X| ≺ |Y |, where |X| is the
length of the word X.
Dealing with ranks, we agree that i − 1 is equal to 1/2 (resp. 0) if
i = 1 (resp. (i = 1/2)), and i + 1 is equal to 1 (resp. 1/2) provided
i = 1/2 (resp. i = 0).
Assuming that the group G(i − 1) for i ≥ 1 is defined, by the period
Ai of rank i we mean the first (relative to the imposed order) of the words
that have infinite order in G(i−1). Then the group G(i) (i ≥ 1) is defined
by imposing the relation An
i = 1 on G(i − 1):
G(i) = 〈a1, a2 | {B = 1, B ∈ T } ∪ {An
1 = 1, . . . , An
i = 1}〉 . (2)
A planar diagram over the presentation (1) (resp. (2)) is called a
diagram of rank 1/2 (resp. i). A cell Π of a diagram ∆ of rank i has
rank 1/2 provided the label of its contour is a cyclic shift of B±1 for some
B ∈ T . Following [7] and [8] any such cell is referred to as 1/2-cell or
a T -cell while cells of rank j, j ≥ 1, are called R-cells. A section q of
a boundary of ∆ is called T -section if φ(q) is freely equal to a subword
of a cyclic shift of B±1 for some B ∈ T . For the definitions of a cell of
rank j ≥ 1 and the strict rank of a diagram we refer to [3]. Similarly to
[3] we define the type τ(∆) of a diagram ∆ of rank i to be the sequence
(ni, . . . , n1/2), where nj is the number of cells of rank j in ∆.
As in [3] we define the concepts of (weak) j-compatibility, j-pair, j-
cell for j ≥ 1. The definition of compatibility of two 1/2-cells (or 1/2-cell
and a T -section) is analogous to the definition of compatibility of two
T -cells (resp. T -cell and an H-section) given in Section 4.2 of [8]. We
only require that φ(t) = 1 in the free group and the label of the path
q1t
−1ptq2 is freely equal to (a subword of ) a cyclic shift of B±1 for some
B ∈ T . Note that a pair of two compatible T -cells (following [3] and [8],
a pair of such cells is called a 1/2-pair) can be substituted by a diagram
without cells.
The definition of a reduced diagram of rank i is preserved. Defining
a strictly reduced diagram of rank i (disk or non-disk) we require in
addition the absence of 1/2-pairs. The definitions of j-bonds and j-
contiguity subdiagrams read the same as in [3]. Notice that now j takes
on values 0, 1/2, 1, 2, . . . . The definitions of a (cyclically) reduced in
rank i and a simple in rank i word are preserved. In the definition of a
smooth section we add one more possibility:
(C3) A T -section s of a contour of a diagram ∆ of rank i is called smooth
provided there are no T -cells in ∆ which are compatible with s.
Such section s is called smooth of rank 1/2. Strictly smooth sections of
rank j are defined for j ≥ 1 only. The definition of a tame diagram need
D. Sonkin 83
not be changed. The definitions of the subgroups F(Aj), G(Aj), K(Aj)
associated with the periods Aj and the definition of F(Aj)-involutions
read the same as in [3].
The formulations and proofs of Lemmas 1.1 - 1.7 from [3] are pre-
served. Lemmas 2.1 - 2.5 together with Lemmas 3.1 - 20.3 are proved by
simultaneous induction on the parameter i, i = 0, 1/2, 1, 2, . . . . Fixing
i and assuming that Lemmas 2.1 - 20.3 hold in rank i − 1, we induct on
the type τ(∆) of a diagram ∆ of rank i.
In Lemmas 2.1 - 2.5 we establish some properties of bonds and con-
tiguity subdiagrams in reduced diagrams of rank i.
Lemma 2.1. Let Γ be a contiguity subdiagram of a cell π to a T -cell
Π in a reduced diagram ∆ of rank i. Then r(Γ) = 0 and the contiguity
degree of π to Π is less than β.
Proof. We prove this lemma by contradiction. Assume that the triple
(π, Γ, Π) is a counterexample with contiguity subdiagram Γ of minimal
type. Let the standard contour of Γ be ∂Γ = d1pd2q, where p = Γ ∧ Π,
q = Γ∧ π. The bonds defining Γ are 0-bonds (otherwise (π, Γ, Π) is not
minimal). That means that |d1| = |d2| = 0. Assuming that Γ has cells,
by Lemma 5.7 (τ(Γ) < τ(∆)) there is a θ-cell. However, by Lemma 2.2
applied to Γ (τ(Γ) < τ(∆)), the degree of contiguity of any cell from Γ
to p is less than β and the contiguity degree of any cell from Γ to q is
less than α by Lemmas 2.5 and 3.4 (again, τ(Γ) < τ(∆)). The fact that
θ > α+β implies that Γ does not have cells. Suppose that the contiguity
degree of π to Π is greater than or equal to β. Lemma 0.1(b) and the
fact that ∆ is reduced mean that π can not be a T -cell. If π is an R-cell,
then we obtain a contradiction with Lemma 0.1(a) since βn2 > 11.
Lemma 2.2. Let Γ be a contiguity subdiagram of a cell π to a T -section
q of the boundary of a reduced diagram ∆ of rank i. Assume there are
no T -cells compatible with q in ∆. Then r(Γ) = 0 and the contiguity
degree of any cell from ∆ to q is less than β (q is a β-section of ∂∆ in
the terminology of [8]).
Proof. The proof is similar to the proof of Lemma 2.1.
Lemma 2.3. Let Γ be a contiguity subdiagram of a T -cell π to a geo-
desic section q in a reduced diagram ∆ of rank i. Then
(1) r(Γ) = 0;
(2) The contiguity degree of π to q does not exceed 1/2 < α.
84 On simple groups of large exponents
Proof. 1. Denote the standard contour of Γ by ∂Γ = d1pd2q1, where
p = Γ ∧ π, q1 = Γ ∧ q. By Lemma 2.1 the bonds defining Γ are 0-bonds,
and therefore |d1| = |d2| = 0. Suppose r(Γ) 6= 0. Then Lemma 5.7 may
be applied to Γ since τ(Γ) < τ(∆), which guarantees that there is a cell
Π ∈ Γ, such that the sum of the contiguity degrees of Π to p and to q1 is
greater than θ. Note that there are no T -cells compatible with the section
q1 in Γ, since otherwise ∆ would not be reduced. Thus, by Lemma 2.2
the degree of contiguity of Π to p is less than β. Lemmas 2.3 and 3.3
applied to Γ (τ(Γ) < τ(∆)) mean that the degree of contiguity of Π to q
is less than α. Finally, the inequality θ > α + β provides a contradiction
to the assumption that Γ has cells.
2. If the statement of part (2) was not true, by proven part (1) the
section q would not be geodesic in ∆.
Lemma 2.4. Let Γ be a contiguity subdiagram of a T -cell π to an R-
cell Π in a reduced diagram ∆ of rank i. Then
(1) r(Γ) = 0;
(2) The contiguity degree of π to Π does not exceed α.
Proof. 1. Denote the standard contour of Γ by ∂Γ = d1pd2q, where
p = Γ ∧ Π, q = Γ ∧ π. By Lemma 2.1 the bonds defining Γ are 0-bonds.
Therefore |d1| = |d2| = 0. Assume that Γ has cells. By Lemma 5.7
applied to Γ (τ(Γ) < τ(∆)) there is a θ-cell in Γ. There are no T -cells
compatible with the section q in Γ since ∆ is reduced. Therefore, by
Lemma 2.2 applied to Γ, the degree of contiguity of any cell from Γ to
q is less than β; by Lemmas 2.5 and 3.4 applied to Γ (τ(Γ) < τ(∆)) the
degree of contiguity of any cell from Γ to p is less than α. Since θ > α+β,
the assumption that Γ has cells is wrong. Part (1) is proved.
2. Assume that the degree of contiguity of a T -cell π to an R-cell Π
of rank j ≥ 1 via Γ is greater than or equal to α. By proven part (1) the
subdiagram Γ does not have cells. This means that there is a common
subpath u of ∂π and ∂Π of length at least α|∂π|. By Lemma 0.2 (α > β)
we conclude that |u| < (1 + ξ)|Aj |. Denote ∂π = uv. If |u| ≤ |Aj |, then
A′ ≡ φ(u)A′′ 1/2
= φ(v)−1A′′ for some cyclic shift A′ of A±1
j .
Consequently, the inequality |v| < α−1(1 − α)|u| < |u| implies that
the word Aj is conjugated in rank 1/2 < j to a shorter word, contrary to
the definition of Aj . Let now |Aj | < |u| < (1 + ξ)|Aj |. Denote u = u1u2,
where |u1| = |Aj |. Then for some cyclic shift A′ of Aj one has
A′ ≡ (φ(u1))
±1 1/2
= (φ(u2v))∓1.
D. Sonkin 85
But
|(φ(u2v))| = |v| + |u2| < (1 − α)α−1|u| + ξ|u1| <
< ((1 − α)(1 + ξ)α−1 + ξ)|u1| < |u1|,
which means that Aj is conjugated in G(1/2) to a shorter word, contrary
to the definition of Aj . Lemma 2.4 is proved.
Lemma 2.5. Let Γ be a contiguity subdiagram of a T -cell π to a smooth
section q in a reduced diagram ∆ of rank i. Then
(1) r(Γ) = 0;
(2) The contiguity degree of π to q does not exceed α.
Proof. The case when q is a smooth section of rank 1/2 is taken care of in
Lemma 2.2. If q is a smooth Aj-periodic section, where Aj is the period
of rank j ≤ i, than the proof is similar to the proof of Lemma 2.4. In
the case when q is a smooth A-periodic section with a simple in rank i
word A the proof also proceeds as in Lemma 2.4 except for instead of the
definition of a period of rank i we now use the definition of a simple in
rank i word.
Below we discuss the changes needed be done in Lemmas 3.1 - 20.3
from [3] for the purpose of present paper. The numeration is preserved.
In the formulation of Lemma 3.1 the section p1 (q1) is considered to
be a smooth T -section of ∂E provided p (q) is either the contour of a cell
Π1 (Π2) of rank j1 = r(Π1) = 1/2 (j2 = r(Π2) = 1/2), or a T -section
of a contour of the diagram ∆. The conclusion of Lemma 3.1 does not
change if none of p1, q1 is a T -section. Otherwise max(|d1|, |d2|) = 0.
Proving Lemma 3.1, we first note that the case when at least one of
p1, q1 is a T -section is taken care of in Lemmas 2.1 and 2.2: the bond E
between p1 and q1 is a 0-bond, and therefore max(|d1|, |d2|) = 0. Now
assume that none of p1, q1 is a T -section and let p1 (q1) be a smooth A-
(B-) periodic section of ∂E. Consider the case when the principal cell π
of the bond E is a T -cell. By Lemmas 2.4, 2.5 r(Γ1) = r(Γ2) = 0, where
Γ1 and Γ2 are the contiguity subdiagrams of π to p1 and q1 respectively,
such that π together with Γ1 and Γ2 form the bond E. Therefore there
are common subpaths of ∂π and each of the sections p1 and q1, and, by
the definition of a bond, the lengths of those subpaths are greater than
β|∂π|. Consequently, by Lemma 0.2, |∂π| < (1 + ξ)β−1 min(|A|, |B|).
Finally,
max(|d1|, |d2|) < |∂π| < 2β−1 min(|A|, |B|) < γ min(n|A|, n|B|).
86 On simple groups of large exponents
In the formulation of Lemma 3.2 we require in addition that l = r(q) >
1/2 (indeed, by Lemmas 2.1, 2.2 the degree of contiguity of any cell to
a T -cell or to a T -section of the boundary is less than β in a reduced
diagram of rank i). In the conclusion of Lemma 3.2 |Ak| is substituted by
n−1|∂Π| since we also have to consider the case r(Π) = 1/2. The proof
of Lemma 3.2 does not change if r(Π) > 1/2. In the case r(Π) = 1/2
Lemma 3.2 is straightforward from Lemmas 2.4, 2.5 and 0.2.
All definitions from Section 4 of [3] are preserved. The formulations
and proofs of Lemmas 4.1 - 4.5 need not be changed. All estimates
from the Lemmas 5.1 - 5.7 hold in view of Lemmas 2.1 - 2.5. In the
proof of Lemma 5.1 notice that r(Π2) 6= 1/2 by Lemma 2.1. If r(Π1) =
1/2, then |qΓ| = |pΓ| since r(Γ) = 0 by Lemma 2.4, and, therefore, the
inequality (5.2) is valid. In the proof of Lemma 5.4, if r(Πk) = 1/2,
where k = 1 and/or 2, then r(Γ) = 0 by Lemma 2.4. Hence |qΓ| = |pΓ|
and the inequality (5.5) follows. Proving Lemma 5.5, notice that case 1
is impossible by Lemma 2.1 if the considered ordinary cell Π is a T -cell.
The formulation of Lemma 6.1 does not change. Proving this lemma
in the case r(q) = 1/2, notice that a diagram ∆′ with contour p′q with
p′ being geodesic does not have cells (otherwise there must be a θ-cell,
but the degree of contiguity of any cell from ∆′ to q is less than β by
Lemma 2.2 applied to ∆′; the degree of contiguity of any cell from ∆′
to p′ is less than α by Lemmas 2.3 and 3.3, but θ > α + β). Then
ρ|q| < |q| = |p′| ≤ |p|.
Now consider the case when q is a smooth B-periodic section, where
B ≡ A±1
l for some l ≤ i or B is a simple word in rank i. The same proof
as in [3] works if rank of the θ-cell Π is greater than 1/2. We need to
consider the situation when the θ-cell Π is of rank 1/2. In the notation
used in the proof of Lemma 6.1 we have |c1| = |c2| = |d1| = |d2| = 0,
v1 = q−1
2 , u1 = p−1
2 by Lemmas 2.3 and 2.5 applied to ∆. The inequality
ρ(|q1| + |q3|) < |p1| + |p3| + (1 − θ)|∂Π| (3)
can be obtained in the same way as in [3].
By Lemma 2.5 the contiguity subdiagram Γq does not have cells. By
Lemma 2.3 applied to the contiguity subdiagram Γp of Π to p and by
the definition of a θ-cell the degree of contiguity of Π to q is greater
than θ − α > β. Consequently, by Lemma 0.2 applied to φ(q2) one has
|q2| < (1 + ξ)|B|.
Assume first that |q2| ≤ |B|. It follows that |q2| ≤ 1/2|∂Π| ≤ |u2| +
|v2|+|p2| since B is cyclically reduced in rank 1/2 and |p2| > θ|∂Π|−|q2| ≥
(θ − 1/2)|∂Π|. Therefore
ρ|q2| < ρ|p2|+ρ(1−θ)|∂Π| < |p2|+((ρ−1)(θ−1/2)+ρ(1−θ))|∂Π| (4)
D. Sonkin 87
Let now |q2| > |B|. Denote q2 = q′q′′, where |q′| = |B|. Then
|q′′| < ξ|q′|. The fact that B is cyclically reduced implies that |q′| ≤
1/2|∂Π| ≤ |q′′|+ |u2|+ |v2|+ |p2|. Therefore |q′| − |q′′| < |u2|+ |v2|+ |p2|.
Consequently,
1 − ξ
1 + ξ
|q2| < (1− ξ)|B| < |q′|− |q′′| < |u2|+ |v2|+ |p2| < |p2|+(1− θ)|∂Π|.
Using the fact that
|p2| > θ|∂Π| − |q′| − |q′′| >
(
θ −
1 + ξ
2
)
|∂Π|,
we obtain
ρ|q2| <
ρ(1 + ξ)
1 − ξ
|p2| +
ρ(1 + ξ)
1 − ξ
(1 − θ)|∂Π| <
< |p2| +
(
(
ρ(1 + ξ)
1 − ξ
− 1
)(
θ −
1 + ξ
2
)
+
ρ(1 + ξ)
1 − ξ
(1 − θ)
)
|∂Π|. (5)
Combining the inequalities (3) and (4) if |q2| ≤ |B|, and (3) and (5)
if |q2| > |B|, we complete the proof in the same way as in [3]. Lemma
6.1 is proved.
The formulations of Lemmas 6.2 - 6.5 do not change. Proving Lemma
6.3 in the case when θ-cell Π is of rank 1/2, note that the bonds defining
the contiguity subdiagrams Γp, Γq are 0-bonds by Lemma 2.1. Analo-
gously, if r(Π) = 1/2 in the proof of Lemma 6.5, then the bonds defining
Γb, Γc are 0-bonds, while the contiguity subdiagrams Γp, Γq do not con-
tain cells by Lemmas 2.3 and 2.5. Thus, proofs of both of these lemmas
in the case r(Π) = 1/2 proceed in the same way as in [3]. We only need to
replace γ by 0 in the estimates and refer to Lemmas 2.5 and 2.3 instead
of Lemmas 3.4 and 3.3.
it is convenient to state Lemma 6.2 as follows.
Lemma 6.2. Let ∆ be a disk tame diagram of rank i. Then |∂Π| ≤
ρ−1|∂∆| for any cell Π ∈ ∆. In particular, if |∂∆| < ρn|Ak| for some
k ≤ i, then r(∆) < k.
All lemmas and estimates of Section 7 from [3] remain valid due to
Lemmas 2.1 − 2.5. However, the presence of T -cells bring the need to
consider several more diagrams of special sort in order to understand the
bond structure of contiguity subdiagrams of rank i. We need to consider
degenerate special 8-gons of rank i and degenerate special 8′-gons of rank
i. The definitions of these new types of diagrams follow.
A disk tame diagram ∆ of rank i is referred to as degenerate special
8-gon of rank i if it possesses the following properties:
88 On simple groups of large exponents
(DG1) The contour ∂∆ of ∆ is considered to be decomposed into the prod-
uct apbrcqds.
(DG2) Each of the sections r and s is either smooth or geodesic.
(DG3) The section p is smooth of some rank k, 1 ≤ k ≤ i; the section q is
smooth of rank 1/2.
(DG4) max(|a|, |b|, |c|, |d|) < 2β−1|Ak|.
(DG5) Let a vertex o1 be chosen on the section p and a vertex o2 be chosen
on the section either (a) r, or (b) s. Denote p(dec, o1) = p1p2 and
(a) r(dec, o2) = r1r2 or (b) s(dec, o2) = s1s2. Suppose that
x = o1 − o2 is a simple path joining the vertices o1 and o2 with
|x| < 2β−1|Ak|. Then the following is true:
1) the subdiagram ∆0 with the contour (a) p2br1x−1 or
(b) p1xs2a contains no cells;
2) the inequality |x| ≤ |b| implies x = b in case (a) or the inequal-
ity |x| ≤ |a| implies x = a−1 in case (b).
(DG6) Let a vertex o3 be chosen on the section q and a vertex o4 be chosen
on the section either (a) r, or (b) s. Denote q(dec, o3) = q1q2 and
(a) r(dec, o4) = r1r2 or (b) s(dec, o4) = s1s2. Suppose that
y = o3 − o4 is a simple path joining the vertices o3 and o4 with
|y| < 2β−1|Ak|. Then the following is true:
1) the subdiagram ∆0 with the contour (a) q1yr2c or
(b) q2ds1y−1 contains no cells;
2) the inequality |y| ≤ |c| implies y = c−1 in case (a) or the
inequality |y| ≤ |d| implies y = d in case (b).
(DG7) There are no bonds between sections r and s in ∆.
A disk tame diagram ∆ of rank i is referred to as degenerate special
8′-gon of rank i if it possesses the following properties:
(DG1′) The contour ∂∆ of ∆ is considered to be decomposed into the prod-
uct apbrcqds.
(DG2′) Each of the sections r and s is either smooth or geodesic.
(DG3′) The section p is smooth of some rank k, 1 ≤ k ≤ i; the section q is
smooth of rank 1/2.
D. Sonkin 89
(DG4′) max(|a|, |b|, |c|, |d|) < 2β−1|Ak|.
(DG5′) There are no bonds between p, r and between p, s in ∆.
(DG6′) There are no bonds between q, r and between q, s in ∆.
(DG7′) There are no bonds between sections r and s in ∆.
The study of degenerate special 8- and 8′-gons proceeds in the same
way as the study of special 8- and 8′-gons in Sections 7, 8 in [3]. Lemma
7.1 remains valid if ∆ is a degenerate special 8-gon of rank i. Deal-
ing with a degenerate special 8-gon of rank i, we consider the same
cases as are considered in [3] for special 8-gon of rank i. Note that
the section q is a β-section by Lemma 2.2. So the case (L4) is impos-
sible. The analog of Lemma 7.2 for degenerate special 8-gon of rank i
says that there are 16 cases to consider: (K1)&(L1), . . . , (K4)&(L3),
(K1)&(L5), . . . , (K4)&(L5). Analog of Lemma 7.3 for degenerate spe-
cial 8-gon of rank i says that the copy of ∆ with decomposition of its
contour b−1p−1a−1s−1d−1q−1c−1r−1 is also a degenerate special 8-gon of
rank i. Thus, the cases (K1)&(L5), . . . , (K4)&(L5) are symmetric to
the cases (K1)&(L2), . . . , (K4)&(L2). Therefore it suffices to consider
12 cases (K1)&(L1), . . . , (K4)&(L3) only. Note that the same cases were
considered in Lemmas 7.4 - 7.15 of [3] studying special 8-gons of rank
i. Analogs of Lemmas 7.4 - 7.15 with some minor changes are valid for
degenerate special 8-gon of rank i.
Thus, using arguments similar to those of Sections 7, 8 from [3] and
Lemmas 2.1 - 2.5 one can prove the following new versions of Lemmas
8.7 and 8.8.
Lemma 8.7. Let ∆ be a special 8-gon of rank i. Then
|p| + |q| < 70β−1M , |r| + |s| < 102β−1M , |∂∆| < 180β−1M ,
where M = max(|Ak|, |Al|).
If ∆ is a degenerate special 8-gon of rank i, then the same estimates
hold with M = |Ak|.
Lemma 8.8. Let ∆ be a special 8′-gon of rank i. Then
|apb| + |cqd| < 100β−1M , |r| + |s| < 130β−1M , |∂∆| < 230β−1M ,
where M = max(|Ak|, |Al|).
For a degenerate special 8′-gon ∆ of rank i the same estimates hold
with M = |Ak|.
90 On simple groups of large exponents
Here is the new version of Lemma 9.1.
Lemma 9.1. Let ∆ be a tame diagram of rank i with the contour
∂∆ = bpcq, where each of the sections p and q is either smooth or geo-
desic. Suppose also that ∆ itself is a contiguity diagram between p and q.
Then there exists a system C(∆) of bonds E1, E2, . . . , Ek between p and
q in ∆ whose standard contours are ∂Et = xtptytqt, where pt = Et ∧ p,
qt = Et ∧ q, t = 1, 2, . . . , k, such that E1, E2, . . . , Ek pairwise have
no cells in common and C(∆) has the following properties:
(a) x1 = b, yk = c and p = p1r1 . . . pk−1rk−1pk, q = qksk−1qk−1 . . . s1q1
for some paths r1, s1, . . . , rk−1, sk−1.
(b) Denote the principal cell of the bond Et by πt, t = 1, 2, . . . , k, (we
set |∂πt| = 0 if Et is a 0-bond) and let the subdiagram ∆t be given
by ∂∆t = y−1
t rtx
−1
t+1st, where t = 1, 2, . . . , k − 1. Then, in this
notation, there are no bonds between the sections rt and st in ∆t
and the following inequalities hold for every t, t = 1, 2, . . . , k−1:
|yt| + |xt+1| ≤ 100β−1n−1 max(|∂πt|, |∂πt+1|) ,
|st| + |rt| ≤ 130β−1n−1 max(|∂πt|, |∂πt+1|) ,
|∂∆t| ≤ 230β−1n−1 max(|∂πt|, |∂πt+1|) ,
provided both πt, πt+1 are R-cells, or
|yt| + |xt+1| ≤ 100β−1n−1|∂π| ,
|st| + |rt| ≤ 130β−1n−1|∂π| ,
|∂∆t| ≤ 230β−1n−1|∂π| ,
where π is the only R-cell among πt, πt+1, or
|yt| <
ξ
10
min(|∂πt|, |∂πt+1|)
if there are no R-cells among πt, πt+1.
In the latter case the subdiagram ∆t does not have cells, |st| = |rt| =
0 and yt = x−1
t+1.
D. Sonkin 91
(c) If r(∆) = j > 0 and Π is a cell of rank j in ∆, then Π is the
principal cell πt of Et for some t, t = 1, 2, . . . , k. In particular,
r(∆) = max
1≤t≤k
(r(πt)) .
Proof. The system of bonds C(∆) is constructed in the same way as in
[3].
Assume that for some t, 1 ≤ t < k the bond Et+1 (or Et) consists of
a T -cell, and the bond Et (resp. Et+1) is either a 0-bond or consists of a
T -cell. We will show that the subdiagram ∆t has no cells.
Assuming that ∆t contains at least one cell, by Lemma 5.7 there is a
θ-cell Π in ∆t. For the sum of the contiguity degrees of Π to the sections
rt, st, we have that (Π, Γr, rt) + (Π, Γs, st) < α + β, since each of the
degrees does not exceed α (following from the fact that p, q are smooth
or geodesic) and if both the degrees of contiguity are at least β we would
have a bond between rt and st.
The sum of the degrees of contiguity of Π to the sections yt and xt+1
does not exceed (Π, Γy, yt) + (Π, Γx, xt+1) < 2β since xt+1 is smooth
of rank 1/2, yt is smooth of rank 1/2 provided Et consists of a T -cell
or |yt| = 0 provided Et is a 0-bond (in the latter case the contiguity
subdiagram Γy does not exist). Thus
θ < (Π, Γr, rt) + (Π, Γs, st) + (Π, Γy, yt) + (Π, Γx, xt+1) < α + 3β,
which is a contradiction. Thus, r(∆t) = 0. The equalities |st| = |rt| = 0
and yt = x−1
t+1 follow now from the fact that there are no bonds between
sections st abd rt. In the case when both Et and Et+1 are 1/2-bonds the
inequality
|yt| <
ξ
10
min(|∂πt|, |∂πt+1|)
follows from Lemma 0.1(b) and the fact that ∆ is reduced.
In view of the above argument, if the subdiagram ∆t contains cells
and if one of the bonds Et, Et+1 consists of a T -cell, then the other bond
is a k-bond for some k ≥ 1. Let Et be a 1/2-bond and Et+1 be a k-bond,
k ≥ 1 (the other case is similar). In the notations from [3], |bt| = |ct| = 0
since r(Γt
p) = r(Γt
q) = 0 by Lemmas 2.3, 2.5. The diagram ∆t with the
decomposition of its contour
∂∆t = a−1
t+1(v
t+1
1 )−1d−1
t+1stv
t
2rt
has properties (DG1′)-(DG7′) of a degenerate special 8′-gon of rank i (if
we set a = a−1
t+1, p = (vt+1
1 )−1, b = d−1
t+1, r = st, |c| = 0, q = vt
2, |d| = 0,
92 On simple groups of large exponents
s = rt). The remaining part of the proof proceeds along the lines of the
proof of Lemma 9.1 given in [3] using the estimates from the new versions
of Lemmas 8.7 and 8.8.
The formulation of Lemma 9.3 is preserved. Proving part (a) in the
case when Et is a 1/2-bond, the estimate |qt| < (1 + ε)|B| is immediate
from Lemmas 2.5 and 0.2 as ξ < ε. If in addition Et+1 is a 1/2-bond,
then by Lemmas 9.1(b) and 0.2 |rt| = |st| = 0 and
|∂∆t| = |yt| + |xt+1| <
ξ
10
min(|∂πt|, |∂πt+1|) <
<
ξ
10
(1 + ξ)β−1|B| < 0.001|B| .
The remaining part of the proof of Lemma 9.3 goes in the same way as
in [3].
The formulation of Lemma 9.4 is preserved. The proof proceeds as
in [3] with some minor changes. The formulation of Lemma 9.5 does not
change. In the case when Π is an R-cell there are no changes in the proof
of 9.5(a). Assume that Π is a T -cell.
First, let Π belong to a subdiagram ∆s, m1 ≤ s ≤ m2 − 1, of
∆(m1, m2). By Lemmas 6.2, 9.1(b) and Lemma 9.5(a) for an R-cell
in ∆(m1, m2)
|∂Π| < 230ρ−1β−1n−1 × 2.22|B| < 2.22|B| .
Let now Π be a principal cell of a bond Et, m1 ≤ t ≤ m2. The bond
Et coincides with the cell Π, and therefore ∂Π = xtptytqt. Applying
Lemma 2.3 (if p is geodesic) or Lemma 2.5 (if p is smooth), one gets
|pt| < α|∂Π|. Therefore, in view of Lemma 9.3(a),
|∂Π| < (1−α)−1(|qt|+ |xt|+ |yt|) < (1−α)−1(1+ε+0.002)|B| < 2.22|B|,
as required.
Lemmas 10.1 - 10.9 are formulated and proved in the same way as in
[3].
The cells Π1, Π2 from the definitions of R- and S-diagrams are now
required to be R-cells. We define T -diagrams of rank i for i ≥ 1 only. The
formulation of Lemma 12.1 is preserved. The proof is decomposed into
cases in the same way as in [3]. Considering the case 1.1.1 in the proof of
Lemma 12.1, the diagram ∆2 is a degenerate special 8′-gon provided the
bond E between r1 and r2 is a 1/2-bond. Therefore the required estimate
can be obtained from Lemma 8.8.
D. Sonkin 93
The investigation of Cases 2.3 - 2.14 of the proof of Lemma 12.1 differs
from the one in [3] if the θ-cell π is a T -cell.
2.3. By Lemma 2.1 the bonds defining Γa do not contain cells. Denote
∂π = uc12vc22, where c1 = c11c12c13, c2 = c21c22c23 and the 0-bonds
E1 and E2 (between π and the sections c1 and c2 respectively), given by
∂E1 = c12c
−1
12 and ∂E2 = c22c
−1
22 , are chosen such that the sum |c13|+|c21|
is minimal.
If there are no bonds between the sections v and q, then, by Lemma
6.5, |v| + |q| ≤ µ(|c13| + |c21|). Therefore
|q| ≤ µ(|c13| + |c21|) < 4µ|Aj | < 0.04n|Aj |.
Now assume that there is a bond between v and q. Consider the
contiguity subdiagram Γ between sections v and q, maximal with respect
to the sum |v2| + |q2|, where q = q1q2q3, v = v1v2v3. By Lemma 2.5(1)
r(Γ) = 0 and therefore q2 = v2. By the choice of Γ, we can apply
Lemma 6.5 to the diagrams ∆1 and ∆2 given by ∂∆1 = c13q1v
−1
1 and
∂∆2 = c21v
−1
3 q3. Hence |q1| + |q3| ≤ µ(|c13| + |c21|) < 4µ|Aj |. In view of
Lemma 0.1(a), |q2| < 11|Aj |. Thus,
|q| = |q1| + |q2| + |q3| < (11 + 4µ)|Aj | < 0.04n|Aj | ,
as required.
Case 2.3 is complete.
2.4. By Lemmas 2.1 and 2.5(1) the bonds between ∂π and each of
the sections r, q are 0-bonds, and the contiguity subdiagrams of π to
the sections r and q are of rank 0. Denote ∂π = r12uq21v, r = r11r12r2,
q = q1q21q22, where r12 (resp. q21) is the maximal common subpath of
∂π and r (resp. q).
Assume that |q21| > β|∂π|. Then by Lemma 0.2 |∂π| < (1+ξ)β−1|Aj |.
Thus, |u| < |∂π| < (1 + ξ)β−1|Aj | < 2β−1|Aj |, giving a contradiction to
the condition (T6).
Let now |q21| ≤ β|∂π|. Then, by Lemma 2.5(2),
|v| > (θ − (α + 2β))|∂π| >
1
3
|∂π|.
Recall also that u < (1− (θ−β))|∂π|. Consider the diagram ∆1 given by
∂∆ = c2r11v
−1q22. Assuming that ∆1 has cells, by Lemma 5.7 there is a
θ-cell Π in ∆1. The contiguity degree of Π to v is less than β by Lemma
2.1 applied to ∆. One of the contiguity degrees of Π to r11 and q22 is
less than β by Lemma 12.1.1. In view of (T6), the arguments from the
proof of Lemma 7.1(c) can be used to show that the sum of contiguity
degrees of Π to r11 andc2, as well as the sum of contiguity degrees of Π
94 On simple groups of large exponents
to q22 andc2, is less than 0.8. Therefore the sum of the contiguity degrees
of Π to all four sections of ∂∆1 is less than 0.8 + 2β, contrary to the
definition of a θ-cell, as θ > 0.8 + 2β. Therefore ∆1 does not have cells,
and |v| ≤ |c2| by the choice of r12 and q21. It follows that |v| ≤ 2β−1|Aj |.
Thus,
1
3
|∂π| < |v| ≤ 2β−1|Aj |.
Consequently, |∂π| < 6β−1|Aj | and
|u| < (1 − (θ − β))|∂π| < 6β−1(1 − (θ − β))|Aj | < 2β−1|Aj | ,
contrary to (T6).
Case 2.4 is complete.
2.5. The bonds defining Γa are 0-bonds. Denote ∂π = c22uc12v,
where c22uc12 = Γa∧∂π, c1 = c11c12c13, c2 = c21c22c23. By (T6) the paths
c11c12 and c22c23 are geodesic in the disk subdiagram of ∆ consisting of
π and Γa. Therefore, in view of Lemma 2.3(1), we may assume that c12
(resp. c22) is the maximal common subpath of ∂π and c1 (resp. c2).
Consider the disk subdiagram ∆1 of ∆ given by ∂∆1 = c23rc11u.
First assume that there is a bond between u and r in ∆1. Let Γ be
a contiguity subdiagram between u and r maximal with respect to the
sum |u2| + |r2|, where u = u1u2u3, r = r1r2r3. The diagram Γ does not
have cells. It follows from Lemmas 2.2, 2.3, 3.3 and 5.7 provided φ(r) is
a reduced in rank i word, or from Lemmas 2.2, 2.5, 3.4 and 5.7 provided
φ(r) is A-periodic with a simple in rank i word A. Let ∆2 be the diagram
given by ∂∆2 = r3c11u
−1
3 . In view of the property (T6) of ∆ one can
estimate the sum of contiguity degrees of a cell from ∆2 to the sections
r3 and c11. Arguing as in the proof of Lemma 7.1(c), and using Lemma
5.7 and the fact that u3 is a β-section of ∂∆2, one can show that ∆2 does
not have cells. Similarly, one gets that ∆3 given by ∂∆2 = u−1
1 c23r1 does
not have cells. Therefore ∂π = c22c23rc11c12v. Lemmas 2.3 and 2.5 imply
that |r| < α|∂π|. Therefore |c22| + |c23| + |c11| + |c12| > (θ − β − α)|∂π|.
Thus,
|v| < (1−(θ−β))|∂π| < (1−(θ−β))(θ−β−α)−1×4β−1|Aj | < 4β−1|Aj |.
If there are no bonds between u and r in ∆1, then, by Lemma 6.5,
|u| < µ(|c11| + |c23|) < 6β−1|Aj | .
Consequently,
|v| < (θ − β)−1(1 − (θ − β))(|c22| + |u| + |c12|) <
D. Sonkin 95
< (θ − β)−1(1 − (θ − β)) × 10β−1|Aj | < 4β−1|Aj |.
Thus, regardless of whether there is a bond between u and r or there
are no such bonds, the length of the path f = c21vc11 in ∆ is estimated
as follows:
|f | = |c21| + |v| + |c11| < 8β−1|Aj |.
Repeating the arguments from [3] following the inequality (12.12), we
obtain the estimate |q| < 0.02n|Aj |.
Case 2.5 is complete. The Case 2.6 is analogous to the Case 2.4, so
we pass to the Case 2.7.
2.7. Consider the following subcases:
(2.7.1.) There is a bond between π, q and there is a bond between π,
c2 in ∆.
(2.7.2.) There are no bonds between π, q and there is a bond between
π, c2 in ∆.
(2.7.3.) There is a bond between π, q and there are no bonds between
π, c2 in ∆.
(2.7.4.) There are no bonds between π, q and between π, c2 in ∆.
2.7.1. The property (T6) of ∆ makes it possible to use arguments
from the proof of Lemma 7.1(c) to show that Γa does not have cells in
this case. Therefore |q| < 11|Aj | by Lemma 0.1(a).
2.7.2. Again referring to the proof of Lemma 7.1(c), we denote ∂π =
uc12vc22r1, where c1 = c11c12c13, c2 = c21c22, r = r1r2. Applying Lemma
6.5 to the disk subdiagram ∆1 given by ∂∆1 = c13qc21v
−1, one gets that
|q| < µ(|c13| + |c21|) < 6β−1|Aj | .
2.7.3. Making use of the arguments from the proof of Lemma 7.1(c),
we denote ∂π = uc12q1vr2, where c1 = c11c12, q = q1q2, r = r1r2r3
and there are no bonds between v, q2, between v, r1 and between v,
c2. By Lemma 0.1(a) |q1| < 11|Aj |. By Lemma 12.1.1 there are no
bonds between r1, q2. Consider the disk subdiagram ∆1 of ∆ given by
∂∆1 = r1v
−1q2c2. Assume that ∆1 has cells. By Lemma 5.7 there is a
θ-cell Π in ∆1. The degree of contiguity of Π to one of the sections r1, q2
is less than β since there are no bonds between those sections. Let this
section be r1 (the case when it is q2 is analogous). Arguing as in Lemma
7.1(c), we get that the sum of contiguity degrees of Π to the sections c2
and q2 is less than 0.8. The section v−1 is a β-section of ∂∆1. So, the
sum of contiguity degrees of Π to all four sections of ∂∆1 is less than
0.8 + 2β. Thus, the inequality θ > 0.8 + 2β means that ∆1 does not
have cells. It follows from the condition (T2) and the fact that there are
no bonds between r1, q2 and between v, q2 that |q2| < 2|Aj | + 2β−1|Aj |.
Finally, |q| = |q1| + |q2| < (13 + 2β−1)|Aj |.
96 On simple groups of large exponents
2.7.4. Denote ∂π = uc12vr2, where r = r1r2r3, c1 = c11c12c13 and
there are no bonds between v, r1 and between v, c13. The disk subdiagram
∆1 of ∆ given by ∂∆1 = c13qc2r1v
−1 has properties (DG1′)-(DG7′) of
a degenerate special 8′-gon of rank i (if we set a = c13, p = q, b = c2,
r = r1, |c| = 0, q = v−1, |d| = |s| = 0). Referring to Lemma 8.8 we get
that |q| < 100β−1|Aj |.
Case 2.7. is complete.
The remaining Cases 2.8 - 2.14 of the proof of Lemma 12.1 with the
θ-cell π of rank 1/2 can be considered in a similar manner.
In the definition of a U -diagram ∆ we require in addition r(∆) = j ≥
1. In the formulation of Lemma 13.1 we add an assumption j = r(∆) >
1/2. Similarly, in Lemmas 13.2, 13.4 we require that j = r(∆(1)) > 1/2
and in Lemma 13.3 - j = r(∆) > 1/2.
The formulation and the proof of Lemma 14.1 do not change. Proving
Lemma 14.2 we need to consider the case r(Π) = 1/2. If pΓ is a subpath of
q, then the needed inequalities are immediate from Lemma 0.2. Since the
bonds defining Γ are 0-bonds, it is possible to obtain the same inequalities
(14.1)-(14.3) modulo substitution 0 for γ. Arguing in the same way as in
[3], instead of (14.4) we get |u| > β|∂Π|.
If there are no bonds between sections y and u in ∆5, then by Lemma
6.5
|y| + |u| ≤ µ(|d1| + |d3|) < µ(1 + δ)|Aj |,
as |d1| = 0, |d3| < (1 + δ)|Aj |. Consequently,
|∂Π| < β−1|u| < β−1µ(1 + δ)|Aj | < 4β−1|Aj |
and
|pΓ| = |y| + |q12| + |t| < (1 + µ)(1 + δ)|Aj | < 5|Aj | .
If there are bonds between y and u in ∆5, then all these bonds are
0-bonds. Since a diagram with contour y′u′ with y′ being smooth and u′
being smooth of rank 1/2 does not have cells, there are decompositions
y = y1y2, u = u2u1, where y1 = u−1
1 is the longest common subpath of y
and u−1. By Lemma 6.5 |y2|+ |u2| ≤ µ|d3| < µ(1+ δ)|Aj |. The fact that
y1 = u−1
1 is a common subpath of a T -section and an Aj-periodic section
means that either |y1| = |u1| < (1 + ξ)|Aj |, or (1 + ξ)|Aj | < |y1| = |u1| <
11|Aj | and |Aj | < ξ−3/2. The latter case is impossible since otherwise
|∂Π| < β−1|u| = β−1(|u1| + |u2|) < β−1(µ(1 + δ) + 11)ξ−3/2 < n2 ,
contrary to Lemma 0.1(3). Therefore
|y|, |u| < (1 + ξ + µ(1 + δ))|Aj |
D. Sonkin 97
and the inequalities sought follow as before.
The formulations and proofs of Lemmas 14.3 - 14.6 do not change.
Proving Lemma 14.7, notice that the inequality 14.19 holds also in the
case when Π is a 1/2-cell. Indeed, in this case when Γ ∧ p1 is a subpath
of q1, r(Γ) = 0 by Lemma 2.5 and, in view of the inequality |pΓ| = |qΓ| >
β|∂Π|, Lemma 0.2 imply the estimate |pΓ| < (1 + ξ)|Aj1 |. If Γ ∧ p1 is
a subpath of t1, then using Lemmas 14.1 and 2.3 we again obtain that
r(Γ) = 0. Therefore |pΓ| ≤ |t1| < δ|Aj1 |. It remains to note that 1+ξ < δ
and the rest of the proof proceeds as in [3]. The formulations and proofs
of Lemmas 14.8 and 14.9 do not change.
Proving Lemma 14.10 we need to consider the case when Π1 is a cell
of rank 1/2. In the notations of [3], if pΓ is a subpath of either q1 or
t1, then r(Γ) = 0 by Lemmas 2.5, 14.1 and 2.3. Therefore, by Lemma
0.1(a) provided pΓ is a subpath of q1 or by Lemma 18.5(c) provided pΓ
is a subpath of t1, |pΓ| < 11|Aj1 |. The obtained estimate implies the
inequality (14.25) from [3].
Now suppose that pΓ is a subpath of p1 but not a subpath of either q1
or t1. By Lemma 2.1 the bonds defining Γ are 0-bonds. By Lemma 6.1
applied to Γ, ρ|qΓ| ≤ |pΓ|. As in the proof of Lemma 14.2 one can obtain
a reduced diagram ∆1 with ∂∆1 = qΓyz, where y is a strictly smooth
Aj1-periodic section, |y| > |pΓ| − (1 + δ)|Aj1 |, and |z| < (1 + δ)|Aj1 |.
Arguing as in the proof of Lemma 14.2, |y| + |qΓ| ≤ µ|z| < µ(1 + δ)|Aj1 |
by Lemma 6.5 provided there are no bonds between sections y and qΓ
in ∆1. It follows that |pΓ| < |y| + (1 + δ)|Aj1 | < (µ + 1)(1 + δ)|Aj1 |.
If there are bonds between y and qΓ in ∆1, then, as in the proof of
Lemma 14.2, we obtain the estimate |y| < (11+µ(1+δ))|Aj1 |. Therefore
|pΓ| < (11 + (µ + 1)(1 + δ))|Aj1 |. In both cases |pΓ| < β−1|Aj1 |, which
forces the inequality (14.25).
If now pΓ is a subpath of a contour of a cell Π2 in ∆, then r(Γ) = 0
by Lemma 2.1. Therefore |pΓ| = |qΓ| and the inequality (14.28) can be
obtained as in [3].
There are no changes in the formulation and in the proof of Lemma
14.11. Note that in the proof the weight of a cell π of rank 1/2 is given
by the formula ν(π) = |∂π|2/3. Lemmas 14.12 - 16.6 are formulated and
proved without changes.
In the formulations of Lemmas 17.1, 17.2 we add the condition r(∆) >
1/2. The conclusion of Lemma 17.2 now reads:
Then ∆0 contains either a 16β−1n−1-contractile cell Π of rank r(Π) >
1/2 or a cell π of rank j ≤ i, j > 1/2 such that if o ∈ ∂π is a phase vertex
and t = o− o is a simple path in ∆0 homotopic as a cyclic path in ∆0 to
a contour of ∆0, then φ(t)
i
= T , where T is an F(Aj)-involution.
98 On simple groups of large exponents
Proof of Lemma 17.2 proceeds in the same way as in [3]. Remark
only that in the sequence of diagrams ∆0 = ∆
(1)
0 , ∆
(2)
0 , . . . the diagram
∆
(k)
0 (for every k > 1) is obtained from ∆
(k−1)
0 by removal of a reducible
j-pair (for some j, 1 ≤ j ≤ i) and all reducible 1/2-pairs.
The formulation of Lemma 17.3 and the scheme of proof of this lemma
in the case r(∆) > 1/2 coincide with those in [3]. The details of the
proof that require additional consideration caused by presence of 1/2-
cells will be considered below. The equality r(∆) = 1/2 means that ∆
consists of 1/2-cells only. Therefore ∆ can be considered as a diagram
over presentation considered in Section 4.2 [8] with additional assumption
that the subgroup N of H is the whole H. It means that Lemma 4.12
from [8] (up to change of notations) may be applied to ∆ (indeed, the
estimates on the lengths of the sections of ∂∆ used in the formulation of
Lemma 4.12 in [8] follow from the analogous estimates in the formulation
of Lemma 17.3 and the fact that ∆ itself is a contiguity subdiagram
between the sections p and q; simplicity of the word A mentioned in
the formulation of Lemma 17.3 implies that in context of [8] the word
A is simple in some positive rank and can be considered as a period of
some rank; in the arguments from [8] (and [7]) that use the fact that the
exponent is odd we refer to Lemma 0.3 instead). Thus, the formulation
of Lemma 17.3 admits the following addition:
If r(∆) = 1/2, then there are two phase vertices op ∈ p, oq ∈ q that
can be joined in ∆ by a path whose label is T
1/2
= 1, and the subgroup
〈A−lTAl | l = 0, 1, 2, 3〉
of G(i) is trivial.
Proving Lemma 17.3 in the case when r(∆) = j > 1/2, as in [3],
by Lemmas 9.4 and 9.5 we find a rigid subdiagram ∆(m1, m2) in ∆.
Note that there might be T -cells in ∆(m1, m2) with boundaries longer
than n|Aj |, however, by Lemma 9.5(a) |∂Π| < 2.22|A| for any cell Π ∈
∆(m1, m2). So, instead of the inequality (17.30) we use two inequalities:
|∂Π| ≤ n|Aj | < 2.22|A|, if r(Π) > 1/2,
|∂Π| < 2.22|A|, if r(Π) = 1/2.
Proving lemma 17.3.1, notice that the cell Π is an R-cell by Lemma
17.2. In the case 4 considered in the proof of Lemma 17.3.1 the subdi-
agram Γt
q contains Π and therefore jt
q = r(Γt
q) > 1/2. The inequality
(17.39) is true for the length of contours of all subdiagrams ∆s(Γ
t
q), and
those of the subdiagrams Es(Γ
t
q) that have an R-cell as its principal cell.
D. Sonkin 99
In the formulation of Lemma 17.3.2 we remark that r(πt′) = j >
1/2, and similarly, r(πm3
) = j > 1/2 in the definition of the diagram
∆(m3, m4) and in the formulation of Lemma 17.3.3. The proofs of these
lemmas are preserved.
The formulation of Lemma 17.3.4 does not change. Proving part (a)
of this lemma, we need to consider the situation when the initial vertex
(pm3
)− of the path pm3
belongs to a subpath p̃t = Ẽt ∧ p̃(m3, m4) for
some t, m3 ≤ t ≤ m4, with the bond Ẽt consisting of a 1/2-cell π̃t. As
in [3], denote p̃t2 to be the subpath of p̃t that connects vertex (pm3
)− to
the terminal vertex (p̃)+ of the path p̃t.
Assume that |p̃t2| < max(β|∂π̃t|, βn|Aj |). Then, using Lemma 9.1(b)
and the inequalities (17.30), the length of the path f1 = p̃t2ỹt is estimated
as follows
|f1| < β max(|∂π̃t|, n|Aj |) + max
( ξ
10
|∂π̃t|, 100β−1|Aj |
)
< 2.5β|A| .
Now assume that |p̃t2| > max(β|∂π̃t|, βn|Aj |) and, as in [3], consider
two cases:
1. (p̃)+ ∈ pm3
.
2. (p̃)+ /∈ pm3
.
1. Consider the diagram Γ1 given by
∂Γ1 = am3
p̃t2h2v.
Here |am3
| < 2β−1|Aj |, |h2| < β−1|Aj | by Lemmas 3.1 and 9.3(b),
and v is an arc of ∂πm3
. By Lemma 6.1 applied to Γ1
|v| ≥ ρ|p̃t2| − |am3
| − |h2| > 0.94βn|Aj | .
In view of |p̃t2| > β|∂π̃t| and Lemma 0.1(b) we can consider p̃t2 as
a T -section of ∂Γ1 and remove (if necessary) 1/2-cells in Γ1 compatible
with p̃t2. Thus we may assume that p̃t2 is a smooth section of rank 1/2
of ∂Γ1.
Let Γ2 be a contiguity subdiagram between v and p̃t2 in Γ1 maximal
with respect to the sum |v2| + |p̃2
t2|, where v2 = Γ2 ∧ v, p̃2
t2 = Γ2 ∧ p̃t2,
v = v1v2v3, p̃t2 = p̃1
t2p̃
2
t2p̃
3
t2. By Lemma 2.2 (applied to the diagram
consisting of Γ1 and πm3
) r(Γ2) = 0. It follows now from Lemma 6.5 and
the choice of Γ2 that
|v1| + |v3| ≤ µ(|am3
| + |h2|) < 5β−1|Aj |.
Then the length of A±1
j -periodic subword φ(p̃2
t2) of φ(p̃t2) can be es-
timated as follows:
|φ(p̃2
t2)| = |φ(v2)| > (0.94βn − 5β−1)|Aj | > 0.9βn|Aj |.
100 On simple groups of large exponents
But this is impossible by Lemma 0.1(a).
2. Consider the subdiagram Γ of ∆(l) consisting of πm3
, π̃t and Γm3
p .
Assume that Γ is reduced. Then, following from rigidity of ∆(m3, m4)
and Lemma 3.4, the contiguity degree of πm3
to π̃t is greater than χ−α >
β. But this contradicts Lemma 2.1. If Γ is not reduced, then the cell π̃t
with some cell from Γm3
p form a 1/2-pair. Removing this 1/2-pair and
taking out the cell πm3
from Γ, we obtain a reduced diagram Γ1 with
∂Γ1 = vam3
p′bm3
. Removing (if necessary) 1/2-cells in Γ1 compatible
with the section p′, we may assume that p′ is a smooth section of rank
1/2 of ∂Γ1. The estimate |v| > 0.94βn|Aj | follows now from rigidity of
∆(m3, m4) and Lemma 3.4. Therefore one can obtain a contradiction to
Lemma 0.1(a) in the same way as it was done in case 1.
To prove part (b) of Lemma 17.3.4 one has to consider the case when
(ỹm4
)− ∈ pt = Et ∧p(m3, m4) with the bond Et consisting of a T -cell Π.
By Lemma 9.3(a) |pt| < (1 + ξ)|A|. The remaining part of the proof of
(b) as well as the proof of (c) proceed as in [3]. Lemma 17.3.4 is proved.
The formulation of Lemma 17.3.5 remains the same. The estimate in
part (a) is valid if the bond E0
k0+1 consists of a 1/2-cell:
Lemma 6.5 applied to ∆0
k0+1 and the inequalities (17.30), (17.63)
imply that
|r0
k0+1| + |s0
k0+1| ≤ µ(|y0
k0+1| + |c0(l)|) < µ(2.22|A| + 0.663|A|) < 4|A|.
Considering case 1 of the proof of Lemma 17.3.5.1 with the addi-
tional condition that the bond E0
1 consists of a 1/2-cell, notice that
the subdiagram ∆0
0 is a degenerate special 8-gon of rank i (if we set
a = d, p = v, b = e, r = r0
0, q = (x0
1)
−1, s = s0
0). Now, by Lemma 8.7,
|∂∆0
0| < 180β−1|Aj |, and therefore r(∆0
0) < j by Lemma 6.2.
Dealing with case 2 of the proof of Lemma 17.3.5.1 in the situation
when r(E0
k0+1) = 1/2, one needs to substitute the summand (1+2γ)n|Aj |
by 2.22|A| in the estimate of the length of ∂∆0
k0+1. The inequality
|∂∆0
k0+1| < 7|A| is still valid and the proof of Lemma 17.3.5.1, as well as
of Lemma 17.3.5, can be completed in the same way as in [3].
The formulations and proofs of Lemmas 17.3.6 - 17.3.8 do not need
any changes. In the formulation of Lemma 17.4 we require in addition
that the rank m of the section φ(q) is greater than 1/2. If the diagram ∆
from the condition of Lemma 17.4 satisfies r(∆) = 1/2, then the subdia-
gram ∆(m1, m2) can be obtained from ∆ = ∆(1, k) by removing those
of the bonds E1, Ek defining ∆, that are not 0-bonds. The inequalities
(17.92)-(17.94) follow from Lemmas 9.3(a) and 9.5.
Proving the existence of a short path connecting the vertex o1
p ∈ pt
with some vertex o1
q ∈ q(m2, m1) in the case when Et = πt is a cell of
D. Sonkin 101
rank 1/2, notice that |pt| < α|∂πt| < 2.22α|Am| by Lemma 2.5 and the
inequality (17.94). Using Lemma 9.3(a) and the inequality (17.93), one
can obtain a path connecting o1
p with some vertex o1
q ∈ q(m2, m1) of
length at most
(2.22α
2
+ 0.003
)
|Am| < 0.6|Am|,
as required. The rest of the proof proceeds as in [3].
The formulation and the proof of Lemma 18.1 need not be changed.
The formulation of Lemma 18.2 is preserved. Instead of the word
B considered in [3] we take the word B̄ obtained from B substituting
every occurrence of the letter a1 by a−1
1 . It follows from the way B is
constructed that the word B̄ is cyclically reduced, |B| = |B̄|, and B̄ does
not contain subwords of the form D3, |D| > 0. Moreover, no cyclic shift
of B̄±1 contains a positive (negative) subword of length greater than 4.
Therefore the maximal length of a common subword of a B̄±1-periodic
word and an element from T is less than 4 < βn2. The remaining part
of the proof proceeds as in [3].
In the formulation of Lemma 18.3 we need to consider only diagrams
∆k of strict rank greater than 1/2. Moreover, we consider only those of di-
agrams ∆k, for which the diagrams Γl
k and Γl
k(m
k
1, mk
2) constructed in the
beginning of the proof of Lemma 18.3 and in Lemma 18.3.1 are of strict
rank greater than 1/2. Indeed, by Lemma 17.3 applied to Γl
k(m
k
1, mk
2) in
the case r(Γl
k(m
k
1, mk
2)) = 1/2 and by construction of Γl
k(m
k
1, mk
2) and
∆l
k, there are phase vertices ōk
1 ∈ pk and ōk
2 ∈ qk and a path rk = ōk
1 − ōk
2
in ∆k such that φ(rk)
1/2
= 1 and the subgroup
〈Al
i+1φ(rk)A−l
i+1 | l = 0, 1, 2, 3〉
of G(i) is trivial.
In the estimate for l in the formulation of Lemma 18.3.1 we substitute
i by i + 1. The length of the chain (18.13) is now bounded from above
by i + 1. The remarks about ranks of diagrams Γl
k(m
k
1, mk
2) mean that
jk = r(πmk
0
) > 1/2 for the cell πmk
0
∈ ∆k(m
k
1, mk
2) from the condition
of Lemma 18.3.2. The proof of Lemma 18.3.1 proceeds as in [3] with
obvious changes caused by new estimate for l. The proof of Lemma
18.3.2 is preserved.
In the condition of Lemma 18.3.3 jk′
= r(π
mk′
0
) > 1/2 by construction
of ∆k′(mk′
1 , mk′
2 ); the proof remains unchanged. Notice that r(∆k(g)) >
1/2 provided claim (2) of Lemma 18.3.3 holds.
The formulations and proofs of Lemmas 18.3.4 - 18.3.6 and the argu-
ment completing the proof of Lemma 18.3 are preserved.
102 On simple groups of large exponents
The formulations of Lemmas 18.4, 18.4.1 and 18.4.1.1 are preserved.
The diagrams constructed in the proof of part (a) of Lemma 18.4 and in
the beginning of the proof of part (b) are of strict rank greater than 1/2
since otherwise, by Lemma 17.3, the word Ai+1 would be of finite order
in G(i) provided Sk is nontrivial in G(i).
In the condition of Lemma 18.4.3 we also allow the section q to be
smooth T -section of length |q| > βn2. The proofs of Lemmas 18.4.2 and
18.4.3 do not change. In the case of a 1/2-cell Π considered in the proof
of Lemma 18.4.2 every letter that occurs in φ(∂Π) occurs also in φ(ut),
since |ut| > β|∂Π| > βn2, the alphabet consists of two letters and the
word φ(∂Π) does not contain long periodic subwords.
The formulation of Lemma 18.5 does not change. In the proof of part
(b) in the case when r(∆) = 1/2 Lemma 17.3 implies that some phase
vertices op ∈ p and oq ∈ q can be connected by a path of zero length.
The formulation of Lemma 19.1 is preserved. Notice that the possi-
bility of the diagram ∆ from the condition of Lemma 19.1 to be of strict
rank 1/2 can be eliminated using similar arguments as in [7] (Lemma
18.9) as the group G(1/2) is torsion free. The same argument can be also
applied to the rigid subdiagram ∆(m1, m2) of ∆. Thus we may assume
that r(∆(m1, m2)) > 1/2. The formulation and proof of Lemma 19.1.1
need not be changed in view of the above remarks. If now the contiguity
subdiagram ∆0 happened to be of strict rank 1/2, then Lemma 17.3 im-
ply the equality (19.16) with F0
i
= 1. The proof of Lemma 19.1 can be
completed as in [3]. Lemmas 19.2 - 19.6 are formulated and proved with-
out any changes. The formulations of Lemmas 20.1-20.3 are preserved.
Proving Lemma 20.1, the diagram ∆(m1, m2) is constructed in the same
way as in the proof of Lemma 17.4. The remaining part of the proof of
Lemma 20.1, as well as proofs of Lemmas 20.2 and 20.3, proceed as in
[3].
The induction is now complete. By GT we denote the inductive limit
of groups G(i), i = 0, 1/2, 1, 2, . . . :
GT = 〈a1, a2 | {B = 1, B ∈ T } ∪ {An
i = 1}∞i=1〉 .
3. Proofs of theorems
Proof of Theorem A. The claims that GT is infinite and satisfies the
identity xn = 1 are deduced from Lemmas 10.4(a) and 18.2 in the same
way as in [3]. The claim about the structure of finite subgroups of GT
follows, as in [3], from Lemma 15.9.
Assuming that the center of GT is nontrivial, instead of the equality
D. Sonkin 103
(21.6) we consider the equality
ZD̄Z−1D̄−1 i
= 1 ,
where D̄ is obtained from the word D used in [3] by substituting every
occurrence of a1 by a−1
1 . Using the remarks made in the proof of Lemma
18.2 one can prove Lemma 21.2 and complete the argument as in [3].
Passing to a subset T ′ of T we may assume that presentation of the
group GT ′ is constructed. Let us show that the kernels of presentations
of groups GT1
and GT2
are different for any two different subsets T1 and
T2 of T . Without loss of generality, there is a word w ∈ T1 \ T2. Thus,
w = 1 in GT1
. Assuming that w = 1 in GT2
we obtain a disk reduced
diagram ∆ of some rank i over the presentation of GT2
with the boundary
label w. The diagram ∆ has cells since w is not equal to the identity in
a free group. The word w is cyclically reduced. By Lemma 18.1 there is
a cell π in ∆ such that the length of a common subpath of ∂π and ∂∆ is
greater than β|∂π|. But this is impossible by Lemma 0.1 and the choice
of w. Therefore w 6= 1 in GT2
, and the kernels of presentations of GT1
and GT2
are different.
Thus, the groups GT1
and GT2
are quotients of a free group F (a1, a2)
over different normal subgroups provided T1 6= T2. Note that there is only
countably many different homomorphisms of a finitely generated group
F (a1, a2) onto a fixed countable group. So we conclude that the set of
pairwise non-isomorphic groups among {GT ′} is of cardinality continuum,
since so is the set of different subsets of T ′ ⊆ T provided T is infinite.
To show the solvability of the word and conjugacy problems in GT ′ in
the case when T ′ is a recursive subset of T , we repeat the arguments from
[3]. In the proof of Lemma 21.1 the θ-cell Π may happen to be a 1/2-cell.
In this situation Lemma 3.1 implies |d1| = |d2| = 0. Case 1 of the proof
of Lemma 21.1 can be considered in the same way as in [3]. In Case 2 the
subdiagram Γ given by ∂Γ = w1u
−1
1 contains cells since r(Γ) = i + 1. By
Lemma 5.7 there is a θ-cell Π′ in Γ. It follows from the choice of Γ that
the contiguity degree of Π′ to w1 is less than α. The contiguity degree
of Π′ to the section u−1
1 is less than β by Lemma 2.1 applied to ∆. The
inequality θ > α + β means that Case 2 is impossible.
Proof of Theorem B. Pick a subset S of T and obtain a new set of
words T ′ in the following way. In every word from S we delete an arbi-
trary occurrence of a letter (a1 or a2). Denote the set thus obtained by S̄
and set T ′ = (T \S)∪S̄. The set T ′ thus obtained has properties similar
to the properties of the set T listed in Lemma 0.1.
More precisely, no cyclic shift of (an inverse of) an element of T ′
contains a B-periodic subword U of length greater than (1+3ξ)|B| unless
104 On simple groups of large exponents
|B| < ξ−3/2 and |U | < 23|B| < 23ξ−3/2; the symmetrized set obtained
from the set T ′ satisfies the small cancellation condition C ′( 3ξ
10); and any
word from T ′ is a positive word of length at least n2 − 1. It follows that
for any set T ′ obtained from the set T in the way described above one
can use the scheme of Sections 1, 2 to construct the group GT ′ .
Now we are ready to make some alterations to the set T . Consider
the set T decomposed into pairs of words: {ui, vi}, i = 1, 2, . . . . For
i = 1, 2, . . . denote by u′
i (resp. v′i) the word obtained by deleting an
arbitrary occurrence of a1 (resp. a2) from ui (resp. vi). For any sequence
α = (αi)
∞
i=1 of 0’s and 1’s the set Tα is constructed as follows: for every i,
i = 1, 2, . . . , the words ui, vi are included in Tα if αi = 0, and the words
u′
i, v′i are included in Tα otherwise. So, every set Tα contains as a subset
exactly one of the pairs {ui, vi} or {u′
i, v′i} for every i, i = 1, 2, . . . . The
above remarks allow us to assume that the groups GTα
are constructed,
and for every sequence α the group GTα
= F/Nα (F = F (a1, a2) is a
free group) satisfies the identity xn = 1. Note that NαNβ = F for any
two different sequences α and β. Indeed, for some index i the subgroup
NαNβ contains the words ui, vi, u′
i and v′i. It follows that a1 = 1 and
a2 = 1 in the quotient F/NαNβ , and therefore NαNβ = F .
Consider a quotient F/Mα of the group GTα
over its maximal proper
normal subgroup. The group F/Mα is simple, and Mα 6= Mβ for any
two different sequences α and β. Indeed, Nα ⊆ Mα for every sequence α.
Therefore MαMβ = F and consequently Mα 6= Mβ since both are proper
subgroups of F .
Thus, the set of different kernels Mα of homomorphisms of a free
group F of rank 2 onto simple groups of exponent n is of cardinality
continuum, and therefore so is the set of pairwise non-isomorphic groups
in the collection {F/Mα}.
Acknowledgements
The author wishes to thank S.V. Ivanov for helpful discussions. The
author is greatly thankful to A.Yu.Ol’shanskii for suggesting the problem,
useful discussions, comments and constant encouragement.
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Contact information
Dmitriy Sonkin Department of Mathematics
Vanderbilt University
1326 Stevenson Center
Nashville, TN 37240
USA
E-Mail: sonkin@math.vanderbilt.edu
Received by the editors: 12.06.2004
and final form in 15.12.2004.
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