Profinite closures of the iterated monodromy groups associated with quadratic polynomials
In this paper we describe the profinite closure of the iterated monodromy groups arising from the arbitrary post-critically finite quadratic polynomial.
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| Цитувати: | Profinite closures of the iterated monodromy groups associated with quadratic polynomials / I. Samoilovych // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 285-304. — Бібліогр.: 8 назв. — англ. |
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irk-123456789-1560272019-06-18T01:31:19Z Profinite closures of the iterated monodromy groups associated with quadratic polynomials Samoilovych, I. In this paper we describe the profinite closure of the iterated monodromy groups arising from the arbitrary post-critically finite quadratic polynomial. 2017 Article Profinite closures of the iterated monodromy groups associated with quadratic polynomials / I. Samoilovych // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 285-304. — Бібліогр.: 8 назв. — англ. 1726-3255 2010 MSC:20E08, 20E18, 37F10. http://dspace.nbuv.gov.ua/handle/123456789/156027 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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In this paper we describe the profinite closure of the iterated monodromy groups arising from the arbitrary post-critically finite quadratic polynomial. |
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Article |
| author |
Samoilovych, I. |
| spellingShingle |
Samoilovych, I. Profinite closures of the iterated monodromy groups associated with quadratic polynomials Algebra and Discrete Mathematics |
| author_facet |
Samoilovych, I. |
| author_sort |
Samoilovych, I. |
| title |
Profinite closures of the iterated monodromy groups associated with quadratic polynomials |
| title_short |
Profinite closures of the iterated monodromy groups associated with quadratic polynomials |
| title_full |
Profinite closures of the iterated monodromy groups associated with quadratic polynomials |
| title_fullStr |
Profinite closures of the iterated monodromy groups associated with quadratic polynomials |
| title_full_unstemmed |
Profinite closures of the iterated monodromy groups associated with quadratic polynomials |
| title_sort |
profinite closures of the iterated monodromy groups associated with quadratic polynomials |
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Інститут прикладної математики і механіки НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/156027 |
| citation_txt |
Profinite closures of the iterated monodromy groups associated with quadratic polynomials / I. Samoilovych // Algebra and Discrete Mathematics. — 2017. — Vol. 23, № 2. — С. 285-304. — Бібліогр.: 8 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT samoilovychi profiniteclosuresoftheiteratedmonodromygroupsassociatedwithquadraticpolynomials |
| first_indexed |
2025-07-14T08:17:08Z |
| last_indexed |
2025-07-14T08:17:08Z |
| _version_ |
1837610090486562816 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 23 (2017). Number 2, pp. 285–304
c© Journal “Algebra and Discrete Mathematics”
Profinite closures of the iterated monodromy
groups associated with quadratic polynomials
Ihor Samoilovych
Communicated by R. I. Grigorchuk
Abstract. In this paper we describe the profinite closure
of the iterated monodromy groups arising from the arbitrary post-
critically finite quadratic polynomial.
Introduction
In this article we consider the iterated monodromy groups arising from
the post-critically finite (i.e. orbit of the critical point is finite) quadratic
polynomial z2 − c and we denote such group as IMG(z2 − c). Group
IMG(z2 − c) could be represented as the automorphism group of the
rooted binary tree (see [1] or [6]). These groups were studied by Bartholdi
and Nekrashevych [1]. They show that properties of such group highly
depend on whether the orbit of the critical point of z2 − c is periodic
(the orbit of the critical point contain critical point itself) or strictly
pre-periodic (the orbit of the critical point do not contain critical point).
Bartholdi and Nekrashevych [1] have proved that group IMG(z2 − c) is
weakly branch on its commutator for the periodic case and also they have
proved that group IMG(z2− c) is regular branch for the pre-periodic case
(except for the infinite dihedral group).
2010 MSC: 20E08, 20E18, 37F10.
Key words and phrases: iterated monodromy groups, quadratic polynomials,
profinite grous, profinite limits, group acting on trees.
286 Profinite closures of monodromy groups
One interesting example of the iterated monodromy group is Basilica
group [5] arising from the complex polynomial z2 − 1. This group is gene-
rated by a three-state automaton. It is the first example of an amenable
group not belonging to the class of sub-exponentially amenable groups [3].
Closure of some iterated monodromy groups of post-critically finite
quadratic polynomials z2 − c was studied by Pink [7]. He studied the
Hausdorff dimension, the maximal abelian factor groups, and the norma-
lizers. Also he showed that the closure of IMG(z2− c) don’t depend up to
conjugacy in AutT on words that define the automorphism group which
is associated with IMG(z2 − c) (we improve this result and we show that
in many cases for different words we get the same closure of IMG(z2 − c),
see Corollary 1 and Theorem 4).
In this article, we provide the description of the closure for the group
IMG(z2− c). We prove that for the pre-periodic case closure of the group
IMG(z2 − c) is a self-similar group of finite type except for the infinite
dihedral group. We use another approach compare to Pink [7] that allow
us to improve some of his results.
1. Preliminaries
Denote by T a rooted binary tree. We can denote vertices of the tree
T as words on the alphabet {0, 1}. The empty word ∅ corresponds to the
root of the tree T ; edges are (v, vx) for all v ∈ T and x ∈ {0, 1}. Words
1 . . . 1︸ ︷︷ ︸
n
and 0 . . . 0︸ ︷︷ ︸
n
we denote by 1n and 0n respectively. The set of all words
of length n we call as n-th level of the tree T and we denote this set by
Tn, n > 0. Let 0Tn denote the set of all words of length n+ 1 that starts
with 0 and let 1Tn denote the set of all words of length n+ 1 that starts
with 1.
The subtree of T of all vertices of the first n levels is denoted by T [n].
The group of all automorphisms of the tree T we denote by AutT . We
denote trivial element in AutT by the symbol 1. For an automorphism
g ∈ AutT and a vertex v ∈ T we denote by g(v) such action on the tree T
that
g(vw) = g(v)g(v)(w)
for all w ∈ T .
A stabilizer of the n-th level of G < AutT , StG(n) (or just St(n)) is
defined to be a set of all elements in G which fix all vertices of the level
n. Group G|T [n] is a restriction of the group G to the subtree T [n] (i.e.
if g ∈ G|T [n] then g ∈ AutT [n] and there exists such h ∈ G that for all
I . Samoilovych 287
v ∈ T [n] equality g(v) = h(v) holds). For the element g ∈ G we denote by
g|T [n] ∈ G|T [n] the restriction of the element g to the subtree T [n]. The
group
G = {g ∈ AutT | g(v)|T [n] ∈ F for every v ∈ T}
called as self-similar group of finite type with depth n given by the pattern
group F < AutT [n]. If self-similar group of finite type with depth n can’t
be given by any pattern group F < AutT [n−1] then n is minimal pattern
depth.
We know [2] that
AutT ∼= ≀∞i=1 Sym({0, 1})
where Sym({0, 1}) = {e, σ} is a symmetric group on {0, 1}. Thus AutT
is a profinite group
AutT = lim←− ≀
n
i=1 Sym({0, 1}).
Also we know [2] that
AutT ∼= AutT ≀ Sym({0, 1})
therefore every element g ∈ AutT we can write by the following wreath
recursion
g = (g(0), g(1))σ
i
where i ∈ {0, 1}. Product of elements g = (g1, g2)σi and h = (h1, h2)σj is
defined as follows
gh =
{
(g1h1, g2h2)σj , if i = 0,
(g1h2, g2h1)σ1−j , otherwise.
To describe profinite closure of the group G it is enough to describe all
G|T [n] for n > 1. In this article we describe G|T [n] for n > 2 recursively i.e.
for g0, g1 ∈ G|T [n−1] and i ∈ {0, 1} we determine necessary and sufficient
condition for g = (g0, g1)σi ∈ G|T [n] .
Let us fix some integer n > 1 and automorphism g = (g0, g1)σi∅ ∈
AutT , where i∅ ∈ {0, 1}. We define following functions
Ln(g) = g(00...00) · g(00...01) · . . . · g(0j2...jn−1jn) · . . . · g(01...11),
Rn(g) = g(10...00) · g(10...01) · . . . · g(1j2...jn−1jn) · . . . · g(11...11),
ln,0(g) = rn,0(g) = l0(g) = r0(g) = i∅,
ln,i(g) = ln,i−1(Ln(g)), for i > 0,
rn,i(g) = rn,i−1(Rn(g)), for i > 0.
288 Profinite closures of monodromy groups
We can define functions ln,i and rn,i for the element g ∈ AutT [k] where
k > ni+ 1 in the same way as above.
We use notation gh for h−1gh where g and h are elements of a group.
For real number x denote by ⌊x⌋ the largest integer not greater than x
and denote by ⌈x⌉ the smallest integer not less than x.
2. Periodic case
We consider profinite closures of iterated monodromy groups associated
with quadratic polynomials with periodic critical points in this section.
Let us fix an integer t > 1 and a word x = x1 . . . xt over the alphabet
{0, 1}. We define elements a1, ..., at ∈ AutT by recursions
a1 =
{
(at, 1)σ, if x1 = 0,
(1, at)σ, otherwise.
and for 2 6 k 6 t
ak =
{
(ak−1, 1), if xk = 0,
(1, ak−1), otherwise.
We will study the group Gt,x = 〈a1, . . . , at〉 and its profinite closure Gt,x.
We consider case x1 = 0 further without limiting the generality because
we can get Gt,1x from Gt,0y for some y by switch of the letters of the
alphabet {0, 1}. Then
a1 = (at, 1).
Note that
a−1
1 = (1, a−1
t )σ
and for 2 6 k 6 t
a−1
k =
{
(a−1
k−1, 1), if xk = 0,
(1, a−1
k−1), otherwise.
Let u be a word in {a1, a2, . . . , at, a
−1
1 , a−1
2 , . . . , a−1
t } that represents ele-
ment g ∈ Gt,x. We denote by |u|k the exponent sum of ak in u. By the
proposition 3.3 from [1] it follows that this numbers does not depend on
word u representing element g ∈ Gt,x. Thus we define |g|k as the exponent
sum of ak for the word u.
I . Samoilovych 289
Lemma 1. For every integer n > 0 we have a2n
k ∈ St(nt + k − 1),
1 6 k 6 t.
Proof. We will prove this by induction on n. Statement for n = 0 follow
from the definition of elements ak, 1 6 k 6 t. We assume that the
statement of the Lemma holds for n− 1 > 0. By the definition of a1 we
have
a2n
1 = (a2
1)2n−1
= (at, at)
2n−1
= (a2n−1
t , a2n−1
t ),
and we see that a2n
1 ∈ St(nt) by the inductive hypothesis. For 2 6
k 6 t the inclusion a2n
k ∈ St(nt + k − 1) follows from the equalities
a2n
k = (a2n
k−1, 1) for xk = 0, a2n
k = (1, a2n
k−1) for xk = 1 and the inclusion
a2n
k−1 ∈ St(nt+ k − 2).
Lemma 2. If g ∈ Gt,x then
∑
v∈0T t−1
|g(v)|1 =
⌈
|g|1
2
⌉
and
∑
v∈1T t−1
|g(v)|1 =
⌊
|g|1
2
⌋
.
Proof. We could represent element from Gt,x as product over the set
{a1, . . . , at, a
−1
1 , . . . , a−1
t }.
We prove the statement by induction on the length of the product. The
base is true as we have the trivial element in that case.
We suppose that the statement is true for h ∈ Gt,x
h = (h0, h1)σi∅ .
We will multiply element from the set {a1, . . . , at, a
−1
1 , . . . , a−1
t } on h. We
denote the result of multiplying by g. There are 2t possibilities for the
element g and we consider all that possibilities. Note that for any v ∈ T t
we have (ai)(v) = 1 or (ai)(v) = ai and for any v ∈ T t we have (a−1
i )(v) = 1
or (a−1
i )(v) = a−1
i .
i) Case g = a1h. Then |g|1 = |h|1 + 1 and
a1h = (ath1, h0)σ1−i∅ .
For the element (a1h)(0) we have the following sum
∑
v∈0T t−1
|(a1h)(v)|1 = |a1|1 +
∑
v∈1T t−1
∣∣∣h(v)
∣∣∣
1
.
290 Profinite closures of monodromy groups
Since the last sum is equal to 1 +
⌊
|h|1
2
⌋
by the inductive hypothesis then
we have
1 +
⌊
|h|1
2
⌋
=
⌈
|h|1 + 1
2
⌉
=
⌈
|g|1
2
⌉
.
For the element (a1h)(1) we have
∑
v∈1T t−1
|(a1h)(v)|1 =
∑
v∈0T t−1
|h(v)|1 =
⌈
|h|1
2
⌉
=
⌊
|h|1 + 1
2
⌋
=
⌊
|g|1
2
⌋
.
ii) Case g = a−1
1 h. Then |g|1 = |h|1 − 1 and
a−1
1 h = (h1, a
−1
t h0)σ1−i∅ .
We have the following sum for the element (a−1
1 h)(0)
∑
v∈0T t−1
|(a−1
1 h)v|1 =
∑
v∈1T t−1
∣∣∣h(v)
∣∣∣
1
.
This number is equal to
⌊
|h|1
2
⌋
by the inductive hypothesis. So we have
∑
v∈1T t−1
∣∣∣h(v)
∣∣∣
1
=
⌊
|h|1
2
⌋
=
⌈
|h|1 − 1
2
⌉
=
⌈
|g|1
2
⌉
.
For the element (a−1
1 h)(1) we have
|a−1
1 |1 +
∑
v∈0T t−1
|h(v)|1 =
⌈
|h|1
2
⌉
− 1 =
⌊
|h|1 − 1
2
⌋
=
⌊
|g|1
2
⌋
.
iii) We consider all cases aih and a−1
i h, 2 6 i 6 t together as for them
we get the same exponent sum of a1: |aih|1 = |a−1
i h|1 = |h|1 and
aih =
{
(ai−1h0, h1)σi∅ , if xi = 0,
(h0, ai−1h1)σi∅ , otherwise.
a−1
i h =
{
(a−1
i−1h0, h1)σi∅ , if xi = 0,
(h0, a
−1
i−1h1)σi∅ , otherwise.
Therefore
∑
v∈0T t−1
|(aih)(v)|1 =
∑
v∈0T t−1
|(a−1
i h)(v)|1 =
∑
v∈0T t−1
|h(v)|1 =
⌈
|h|1
2
⌉
,
∑
v∈1T t−1
|(aih)(v)|1 =
∑
v∈1T t−1
|(a−1
i h)(v)|1 =
∑
v∈1T t−1
|h(v)|1 =
⌊
|h|1
2
⌋
.
I . Samoilovych 291
We prove that functions lt,i and rt,i depend only on the exponent sum
of a1.
Lemma 3. For every element g ∈ Gt,x and for every not negative integer
i the following equalities hold
lt,i(g) =
⌈
|g|1
2i
⌉
mod 2 and rt,i(g) =
⌊
|g|1
2i
⌋
mod 2.
Proof. We prove this Lemma by induction on i. In the case i = 0 we have
lt,0(g) = rt,0(g) = |g|1 mod 2. Now we suppose that the lemma holds for
i− 1 where i > 1. By the definition of lt,i we have
lt,i(g) = lt,i−1(Lt(g)) =
⌈
|Lt(g)|1
2i−1
⌉
mod 2 =
⌈∑
v∈0T t−1 |g(v)|1
2i−1
⌉
mod 2.
We conclude by the Lemma 2 that
lt,i(g) =
⌈
⌈|g|1/2⌉
2i−1
⌉
mod 2 =
⌈
|g|1
2i
⌉
mod 2.
Analogously, for the function rt,i we have
rt,i(g) = rt,i−1(Rt(g)) =
⌊
|Rt(g)|1
2i−1
⌋
mod 2 =
⌊
⌊|g|1/2⌋
2i−1
⌋
mod 2
=
⌊
|g|1
2i
⌋
mod 2.
Theorem 1. For every g ∈ Gt,x and every not negative integer k the
following holds
k∑
i=0
lt,i(g)2i +
k∑
i=0
rt,i(g)2i ≡ 0 mod 2k+1. (1)
Proof. It follows from the Lemma 3 that
k∑
i=0
lt,i(g)2i =
k∑
i=0
(⌈
|g|1
2i
⌉
mod 2
)
2i,
k∑
i=0
rt,i(g)2i =
k∑
i=0
(⌊
|g|1
2i
⌋
mod 2
)
2i.
This equalities are the 2−adic representations for −|g|1 mod 2k+1 and
|g|1 mod 2k+1 respectively.
292 Profinite closures of monodromy groups
Remark 1. Equality 1 also holds for every automorphism g ∈ Gt,x|T [n]
where n > kt+ 1. We can choose such element g′ ∈ Gt,x that g′|T [n] = g.
Equality 1 holds for g′ and so it holds for g.
Lemma 4. If for the automorphism g = (g0, g1)σi∅ ∈ AutT where
g0, g1 ∈ Gt,x, i∅ ∈ {0, 1} and for the integer k > 0 following congruence
relation holds
k∑
i=0
lt,i(g)2i +
k∑
i=0
rt,i(g)2i ≡ 0 mod 2k+1,
then |g1|t + i∅ ≡ |g0|t mod 2k.
Proof. At first we make the following transformation
k∑
i=0
lt,i(g)2i +
k∑
i=0
rt,i(g)2i = 2
k∑
i=1
lt,i(g)2i−1 + 2
k∑
i=1
rt,i(g)2i−1 + 2i∅
≡ 0 mod 2k+1.
Then we divide last congruence by 2 and we get
k−1∑
i=0
lt,i+1(g)2i +
k−1∑
i=0
rt,i+1(g)2i + i∅ ≡ 0 mod 2k.
By the definition of lt,i+1 and rt,i+1 we have
k−1∑
i=0
lt,i+1(g)2i +
k−1∑
i=0
rt,i+1(g)2i =
k−1∑
i=0
lt,i(Lt(g))2i +
k−1∑
i=0
rt,i(Rt(g))2i.
As g0, g1 ∈ Gt,x then g(v) ∈ Gt,x for all non-empty v and we can use
Lemma 3 for the previous expression and we get
k−1∑
i=0
(⌈
|Lt(g)|1
2i
⌉
mod 2
)
2i +
k−1∑
i=0
(⌊
|Rt(g)|1
2i
⌋
mod 2
)
2i
=
k−1∑
i=0
∑
v∈0T t−1
|g(v)|1
2i
mod 2
2i +
k−1∑
i=0
∑
v∈1T t−1
|g(v)|1
2i
mod 2
2i.
From the definition of generators of Gt,x we have
∑
v∈T t−1
|(g0)(v)|1 = |g0|t and
∑
v∈T t−1
|(g1)(v)|1 = |g1|t.
I . Samoilovych 293
Now we have the condition
k−1∑
i=0
(⌈
|g0|t
2i
⌉
mod 2
)
2i +
k−1∑
i=0
(⌊
|g1|t
2i
⌋
mod 2
)
2i + i∅ ≡ 0 mod 2k.
Congruence above contains 2-adic representations for −|g0|t mod 2k and
|g1|t mod 2k respectively which finally lead us to
−|g0|t + |g1|t + i∅ ≡ 0 mod 2k.
Theorem 2. We assume that g = (g0, g1)σi∅ ∈ Gt,x|T [n] and g′ =
(g′
0, g
′
1)σi∅ where g′
0, g
′
1 ∈ Gt,x|T [n], g′
0|T [n−1] = g0 and g′
1|T [n−1] = g1 for
n > 1. Then g′ ∈ Gt,x|T [n+1] if and only if the following congruence holds
for k =
⌊
n
t
⌋
k∑
i=0
lt,i(g
′)2i +
k∑
i=0
rt,i(g
′)2i ≡ 0 mod 2k+1. (2)
Proof. Necessary condition follow from the Theorem 1.
Here we prove the sufficient condition. We can choose such elements
w0, w1 ∈ Gt,x that w0|T [n] = g′
0 and w1|T [n] = g′
1. By the Lemma 4 there
exists such integer j that |w1|t + i∅ = |w0|t + j2k. By the Lemma 1
we have a2k
t ∈ St(kt + t − 1). As n 6 kt + t − 1 then a2k
t |T [n] is trivial
and w0a
j2k
t |T [n] = g′
0. If i∅ = 0 then |w1|t = |w0|t + j2k. Stabilizer St(1)
is generated by a2
1 = (at, at), a
−xi
1 aia
xi
1 = (ai−1, 1) and axi−1
1 aia
1−xi
1 =
(1, ai−1) where 2 6 i 6 t. So we have
(w0a
j2k
t , w1) = (a
|w1|t
t , w1)(a
−|w1|t
t , a
−|w1|t
t )(w0a
j2k
t , a
|w1|t
t ) ∈ St(1).
If i∅ = 1 then we can generate element a1 · (w0a
j2k
t , w1)σ = (atw1, w0a
j2k
t )
because of 1 + |w1|t = |w0|t + j2k.
Remark 2. The congruence 2 from Theorem 2 gives us a restriction only
in the case n = kt for some positive integer k. In other cases congruence 2
holds because kt < n therefore action of g′ on the level n+ 1 not used for
the congruence 2.
Remark 3. We do not use order in products from definitions of Ln(g)
and Rn(g).
294 Profinite closures of monodromy groups
Remark 4. We prove our formula for the case x1 = 0 only. But if
we switch letters in the alphabet {0, 1} then by the previous Remark
we conclude that it is enough to switch functions lt,i and rt,i in the
congruence 2 of the Theorem 2 to get the criteria for Gt,x|T [n] for the case
x1 = 1. But then we get the same criteria as for the case x1 = 0.
Pink [7] already showed that group Gt,x depend only on t up to
conjugacy in AutT . The following corollary improves that result for the
periodic case.
Corollary 1. For all words x1, x2 ∈ {0, 1}t equality Gt,x1 = Gt,x2 holds.
Proof. Any group Gt,x is Sym{0, 1} on the first level. Congruence 2 do
not depend on x therefore Gt,x do not depend on the word x.
Remark 5. In general group Gt,x depend on a word x. For example
when t = 3 groups corresponding to “Douady Rabbit” and “Airplane” [6]
are not isomorphic.
3. Strictly pre-periodic case
In this section, we consider profinite closures of iterated monodromy
groups associated with quadratic polynomials with pre-periodic critical
points.
Let us fix integers t > s > 1 and a word x = x1x2 . . . xt−1 over the
alphabet {0, 1}. We define elements a1, ..., at ∈ AutT by recursions
a1 = (1, 1)σ,
as+1 =
{
(as, at), if xs = 0,
(at, as), otherwise,
ai =
{
(ai−1, 1), if xi−1 = 0,
(1, ai−1), otherwise.
We will study group Gs,t,x = 〈a1, . . . , at〉 and its profinite closure Gs,t,x.
Almost all groups Gs,t,x are self-similar groups of finite type, only one
exception is the case s = 1 and t = 2. We consider this exception later.
Lemma 5. (Proposition 4.2 from[1]) For all 1 6 i 6 t equality a2
i = 1
holds. Abealinization Gs,t,x/G
′
s,t,x is (Z/(2Z))t generated by the images of
the ai.
I . Samoilovych 295
Let u be a word in {a1, a2, . . . , at} that represents element g ∈ Gs,t,x.
We denote by |u|k the exponent sum of ak in u. By the Lemma 5 it follows
that number |u|k mod 2 does not depend on the word u representing
element g ∈ Gs,t,x. Thus we define |g|k as |u|k mod 2.
Lemma 6. Element g ∈ Gs,t,x belong to the commutator G′
s,t,x iff |g|i = 0
for all 1 6 i 6 t.
Proof. Statement follow from the Lemma 5.
Lemma 7. If g ∈ St(i) then |g|i = 0 for all 1 6 i 6 t.
Proof. We prove this by induction on i. If g ∈ St(1) then |g|1 = 0. Let
g = (g1, g2). If g ∈ St(i) then g1, g2 ∈ St(i− 1) and by induction suppose
for g1 and g2 we have that numbers |g1|i−1 and |g2|i−1 are zeroes. Note
that equality |g|i = (|g1|i−1 + |g2|i−1) mod 2 holds therefore |g|i = 0.
Lemma 8. If g ∈ St(t) then g ∈ G′
s,t,x.
Proof. Statement follow from the Lemma 6 and the Lemma 7.
Theorem 3. Suppose that s 6= 1 and t 6= 2 simultaneously. Then group
Gs,t,x is a self-similar group of finite type with the following minimal
depth
d =
{
5, if s = 2 and t = 3,
t+ 1, otherwise.
Proof. We use Sunic’s [8] criteria from the Theorem 3 for a self-similar
group of finite type. It is already proven that the group Gs,t,x is regular
branch over some group Hs,t,x (Theorem 4.10 from [1]). It is only left
to show that Hs,t,x contain a stabilizer of the level d. Let us to consider
following cases.
• If s > 2 and t > s+ 2 or if s > 3 and t = s+ 1 than Hs,t,x = G′
s,t,x
and statement follow from the Lemma 8.
• If s = 1 and t > 3 then Hs,t,x = 〈[ai, aj ]26i<j6t, [a1, aj ]16j<t〉
Gs,t,x .
Group G′
s,t,x/Hs,t,x
∼= Z2 generated by [a1, at]. Therefore
G′
s,t,x = Hs,t,x ∪ ([a1, at]Hs,t,x).
By the Lemma 8 we have StGs,t,x
(t) < G′
s,t,x. We prove that Hs,t,x contain
a stabilizer of the level t in the following way: for any element g ∈ G′
s,t,x
we prove that if g /∈ Hs,t,x then g /∈ StGs,t,x
(t).
296 Profinite closures of monodromy groups
It is easy to see that for all h = (h1, h2) ∈ Hs,t,x we have |h1|t−1 =
|h2|t−1 = 0. As [a1, at]=(at−1, at−1) then for all g=(g1, g2) ∈ [a1, at]Hs,t,x
we conclude that |g1|t−1 and |g2|t−1 are ones; but by the Lemma 7 it
follow that g1, g2 /∈ StGs,t,x
(t− 1). Therefore g /∈ StGs,t,x
(t).
• Case s = 2, t = 3. We consider case x1 = 0 only without limiting
the generality. In this case we have H2,3,x = 〈[a1, a2a3]〉G2,3,x . Also
G′
2,3,x = H2,3,x ∪ ([a1, a2]H2,3,x) ∪ ([a2, a3]H2,3,x)
∪ ([a1, a2][a2, a3]H2,3,x).
Note that St(4) < G′
2,3,x. We prove for any g ∈ G′
2,3,x that if g /∈ H2,3,x
then g /∈ St(4) using GAP in the following way: we compute group
H2,3,x|T [4] and we check that e /∈ ([a1, a2]H2,3,x)|T [4] ∪ ([a2, a3]H2,3,x)|T [4] ∪
([a1, a2][a2, a3]H2,3,x)|T [4] .
Minimality for all cases except s = 2 and t = 3 follow from the fact
that Gs,t,x|T [t] = Aut(T t). For the case s = 2 and t = 3 it can be seen
using GAP.
Group Gs,t,x barely depend on the word x. Pink [7] proved that for
all words z, y ∈ {0, 1}t−1 there exists such w ∈ AutT that Gs,t,z =
wGs,t,yw
−1. The following theorem improves that result.
Theorem 4. Suppose that s 6= 1 and t 6= 2 simultaneously.
• If s > 2 and t > s+2 or if s > 3 and t = s+1. Then for all possible
words y = y1 . . . yt−1 we have Gs,t,x = Gs,t,y.
• If s = 1 and t > 3 then for all possible words y = y1 . . . yt−1 where
xt−1 = yt−1 we have Gs,t,x = Gs,t,y.
• If s = 2, t = 3 then all 4 groups G2,3,x for this case are conjugate.
Proof. All groups in this theorem are self-similar groups of finite type
therefore it is enough to prove the statement for pattern groups.
For the case s = 2, t = 3 using GAP we can see that there are
4 different groups G2,3,x as sets. Pink [7] already prove that all G2,3,x
are conjugate. But we show some example of conjugator for G2,3,00 and
G2,3,01.
We find using GAP such elements f1, f2 ∈ AutT [5] and g1, g2, g3, g4 ∈
G2,3,00|T [4] that
f1 = (g1f2|T [4] , g2f2|T [4]) and f2 = (g3f1|T [4] , g4f1|T [4]),
G2,3,00|T [5] = f1G2,3,01|T [5]f−1
1 and G2,3,00|T [5] = f2G2,3,01|T [5]f−1
2 .
I . Samoilovych 297
Then we can choose such elements ĝ1, ĝ2, ĝ3, ĝ4 ∈ G2,3,00 that ĝi|T [4] = gi.
Therefore we can define elements
f̂1 = (ĝ1f̂2, ĝ2f̂2) and f̂2 = (ĝ3f̂1, ĝ4f̂1).
For ĥ = (h1, h2)σi ∈ G2,3,01, i ∈ {0, 1} the following holds
f̂1(h1, h2)σif̂1
−1
= (ĝ1f̂2h1f̂2
−1
ĝ1+i
−1, ĝ2f̂2h2f̂2
−1
ĝ2−i
−1)σi,
f̂2(h1, h2)σif̂2
−1
= (ĝ3f̂1h1f̂1
−1
ĝ3+i
−1, ĝ4f̂1h2f̂1
−1
ĝ4−i
−1)σi.
As G2,3,00 is self-similar group of finite type then it follows that
f̂1G2,3,01f̂1
−1
= G2,3,00 and f̂2G2,3,01f̂2
−1
= G2,3,00.
One possible choice for ĝi is
ĝ1 = 1, ĝ2 = a2a
3
3a1, ĝ3 = a2
3a2, ĝ4 = a3a2a
2
3a2a1.
We exclude case s = 2, t = 3 from further consideration.
Let us to fix the following elements
a′
s+1 =
{
(a′
s, a
′
t), if ys = 0,
(a′
t, a
′
s), otherwise.
a′
i =
{
(a′
i−1, 1), if yi−1 = 0,
(1, a′
i−1), otherwise.
In our proof we use a notation for the Kronecker delta
δij =
{
0, if i = j,
1, if i 6= j.
Then we have for 2 6 i 6 t:
ai = a
xi−1
1 (ai−1, a
δi,s+1
t )a
xi−1
1 ,
a′
i = a
yi−1
1 (a′
i−1, (a
′
t)
δi,s+1)a
yi−1
1 .
We define function φ on the set of all words in the alphabet {a1, . . . , at−1}
φ : {a1, . . . , at−1}
∗ → {a1, . . . , at}
∗
298 Profinite closures of monodromy groups
recursively as follows
φ(∅) = ∅,
φ(ai) = axi
1 ai+1a
xi
1 , 1 6 i 6 t− 1,
φ(αβ) = φ(α)φ(β), where α, β ∈ {a1, . . . , at−1}
∗.
Note that a0
i is the empty word therefore φ(a0
i ) = 1.
We define function ψ(w) as element from the group Gs,t,x that corre-
spond to the word w ∈ {a1, . . . , at}
∗, note that ψ(∅) = 1. Functions ψ
and φ are homomorphisms. Let us to show that the following properties
hold for all words w ∈ {a1, . . . , at−1}
∗
ψ(φ(w)) = (ψ(w), a
|w|s
t ), |ψ(φ(w))|1 = 0.
We prove these by induction on length of the word. For the first one we
have
ψ(φ(∅)) = ψ(∅) = 1 = (ψ(∅), 1),
ψ(φ(ai)) = ψ(axi
1 ai+1a
xi
1 ) = axi
1 ai+1a
xi
1
= axi
1 a
xi
1 (ai, a
δi+1,s+1
t )axi
1 a
xi
1 = (ai, a
|ai|s
t ),
ψ(φ(aiw)) = ψ(φ(ai)φ(w)) = ψ(φ(ai))ψ(φ(w)) = (aiψ(w), a
|ai|s+|w|s
t ).
For the second one we have
|ψ(φ(∅))|1 = |1|1 = 0,
|ψ(φ(ai))|1 = |axi
1 ai+1a
xi
1 |1 = 0, 1 6 i 6 t− 1,
|ψ(φ(aiw))|1 = |ψ(φ(ai))|1 + |ψ(φ(w))|1 = 0.
Without limiting the generality it is enough to show that
〈a′
1, . . . , a
′
t〉|T [t+1] < 〈a1, . . . , at〉|T [t+1] .
We construct such elements from Gs,t,x that act on first t+ 1 levels as a′
i.
So we define
c1 = ∅, ci = a
yi−1
1 φ(ci−1)a
xi−1
1 , 2 6 i 6 t,
d1 = a′
1 = a1, di = ψ(ci)aiψ(ci)
−1, 2 6 i 6 t.
Note that ci ∈ {a1, . . . , at−1}
∗ for 1 6 i 6 t . We prove the following
di = a
yi−1
1 (di−1, a
δi,s+1
t )a
yi−1
1
I . Samoilovych 299
by induction on i.
c2 = ay1
1 a
x1
1 ,
d2 = ay1
1 a
x1
1 a2a
x1
1 a
y1
1 = ay1
1 (a1, a
δ2,s+1
t )ay1
1 .
di = ψ(ci)aiψ(ci)
−1 = a
yi−1
1 ψ(φ(ci−1))a
xi−1
1 ai(a
yi−1
1 ψ(φ(ci−1))a
xi−1
1 )−1
= a
yi−1
1 (ψ(ci−1)ai−1ψ(ci−1)−1, a
δi,s+1
t )a
yi−1
1 = a
yi−1
1 (di−1, a
δi,s+1
t )a
yi−1
1 .
We have ds+1 = ays
1 (ds, at)a
ys
1 . But we need an element
ds+1 = ays
1 (ds, dt)a
ys
1 ∈ Gs,t,x.
We construct this element for each case.
• Case s > 2 and t > s+ 2 or if s > 3 and t = s+ 1. There exist such
b1 ∈ 〈a1, a2〉 and b2 ∈ 〈a1, a2, a3〉 that [ds, a
b1
t ] = 1 and b2 = (b1, 1).
If s > 2 and t > s+ 2 then
b1 = a
xt−1+ys−1+1
1 .
If s > 3 and t = s+ 1 then
b1 = a
xt−1
1 ψ(φ(a
(xt−2+ys−2+1) mod 2
1 ))a
ys−1
1 .
For both cases
b2 = ψ(φ(b1)).
Then we compute the following
b3 = ays
1 ψ(φ(ct))
a1b2ays
1 = ays
1 ((ar
t )b1 , ψ(ct))a
ys
1 , r = |ψ(ct)|s,
ds+1 = b3ds+1b
−1
3 = b3(ays
1 (ds, at)a
ys
1 )b−1
3
= ays
1 ((ar
t )b1ds((ar
t )b1)−1, ψ(ct)atψ(ct)
−1)ays
1 = ays
1 (ds, dt)a
ys
1 .
• Case s = 1 and t > 3 and xt−1 = yt−1. Therefore
|ψ(ct)|1 = |ψ(a
yt−1
1 φ(ct−1)a
xt−1
1 )|1 = (yt−1 + xt−1) mod 2 = 0,
b3 = ays
1 ψ(φ(ct))
a1ays
1 = ays
1 (a
|ct|1
t , ψ(ct))a
ys
1 = ays
1 (1, ψ(ct))a
ys
1 ,
ds+1 = b3ds+1b
−1
3 = b3(ays
1 (ds, at)a
ys
1 )b−1
3
= ays
1 ((ds, ψ(ct)atψ(ct)
−1)ays
1 = ays
1 (ds, dt)a
ys
1 .
300 Profinite closures of monodromy groups
We prove that di|T [t] = a′
i|T [t] for 1 6 i 6 t by induction on i. Base is true
because d1 = a′
1. We have for 2 6 i 6 t, i 6= s+ 1 the following
di|T [t] = a
yi−1
1 |T [t](di−1|T [t−1] , 1|T [t−1])a
yi−1
1 |T [t]
= a
yi−1
1 |T [t](a′
i−1|T [t−1] , 1|T [t−1])a
yi−1
1 |T [t] = a′
i|T [t] ,
ds+1|T [t] = ays
1 |T [t](ds|T [t−1] , at|T [t−1])a
ys
1 |T [t]
= ays
1 |T [t](a′
s|T [t−1] , 1|T [t−1])a
ys
1 |T [t] = a′
s+1|T [t] .
Therefore
di|T [t+1] = a
yi−1
1 |T [t+1](di−1|T [t] , 1|T [t])a
yi−1
1 |T [t+1]
= a
yi−1
1 |T [t+1](a′
i−1|T [t] , 1|T [t])a
yi−1
1 |T [t+1] = a′
i|T [t+1] ,
ds+1|T [t+1] = ays
1 |T [t+1](ds|T [t] , dt|T [t])a
ys
1 |T [t+1]
= ays
1 |T [t+1](a′
s|T [t] , a′
t|T [t])a
ys
1 |T [t+1] = a′
s+1|T [t+1] .
So 〈a′
1, . . . , a
′
t〉|T [t+1] < 〈a1, . . . , at〉|T [t+1] .
3.1. Case s = 1 and t = 2
Here we consider the profinite closureG = G1,2,0 of the infinite dihedral
group G = G1,2,0 generated by following automorphisms
a1 = (1, 1)σ, a2 = (a1, a2).
Note that a−1
1 = a1 and a−1
2 = a2. Let us to compute a1a2 and a2a1
a1a2 = (a2, a1)σ, a2a1 = (a1, a2)σ.
Then for all integer k we have
(a1a2)k = (a1a2)2⌊ k
2⌋+k mod 2 = ((a1a2)2)⌊
k
2⌋(a1a2)k mod 2
= (a2a1, a1a2)⌊
k
2⌋(ak mod 2
2 , ak mod 2
1 )σk mod 2.
Therefore
(a1a2)k = ((a1a2)−⌊ k
2⌋ak mod 2
2 , (a1a2)⌊
k
2⌋ak mod 2
1 )σk mod 2. (3)
Lemma 9. For all integer n > 0 the following equality holds
∏
v∈T n
l0(((a1a2)k)(v)) =
{
1, if 2n | k and 2n+1 ∤ k,
0, otherwise.
I . Samoilovych 301
Proof. Let us prove this by induction on n. One can directly check the
equality for n = 0, 1. We assume that Lemma holds for n− 1 where n > 2.
If 2n | k and 2n+1 ∤ k then k = k02n for some odd k0 and by equality 3
(a1a2)k = ((a1a2)−k02n−1
, (a1a2)k02n−1
).
So in this case we have
∏
v∈T n
l0(((a1a2)k)(v))
=
∏
v∈T n−1
l0(((a1a2)−k02n−1
)(v)) ·
∏
v∈T n−1
l0(((a1a2)k02n−1
)(v)) = 1 · 1 = 1.
Let us to consider other cases. As n > 2 and a1 act trivially on all levels
except first we have for all v ∈ Tn−1
l0(((a1a2)k)(1v)) = l0(((a1a2)⌊
k
2⌋ak mod 2
1 )(v)) = l0(((a1a2)⌊
k
2⌋)(v)).
If
⌊
k
2
⌋
6= k02n−1 for all odd k0 then
∏
v∈T n
l0(((a1a2)k)(v)) =
∏
v∈0T n−1
l0(((a1a2)k)(v)) ·
∏
v∈T n−1
l0(((a1a2)⌊
k
2⌋)(v))
=
∏
v∈0T n−1
l0(((a1a2)k)(v)) · 0 = 0.
If
⌊
k
2
⌋
= k02n−1 for some odd integer k0 then k = k02n + 1 (because now
we consider k 6= k02n)
l0(((a1a2)k)(0v)) = l0(((a1a2)−k02n−1
a2)(v))= l0(((a1a2)−k02n−1
a2a1a1)(v))
= l0(((a1a2)−k02n−1−1a1)(v)) = l0(((a1a2)−k02n−1−1)(v)).
Then
∏
v∈T n
l0(((a1a2)k)(v))
=
∏
v∈T n−1
l0(((a1a2)−k02n−1−1)(v)) ·
∏
v∈1T n−1
l0(((a1a2)k)(v)) = 0.
Lemma 10. For all g ∈ G and all integer n > 2 the following congruence
holds
l0(g(01n−1)) + l0(g(1n)) +
∏
v∈T n−1
l0(g(v)) ≡ 0 mod 2. (4)
302 Profinite closures of monodromy groups
Proof. Every element of the group G is the product (a1a2)kak1
1 where k
is some integer and k1 ∈ {0, 1}. As a1 act not trivially on the first level
only it is enough to prove congruence for (a1a2)k for all integer k.
We use the following notations: for every integer n > 0 and every
integer k we denote
αk
n+1 = l0(((a1a2)k)(01n)),
βk
n = l0(((a1a2)k)(1n)).
From the equality 3 we conclude that for all n > 1 and all integer k
βk
0 = k mod 2,
αk
n = l0(((a1a2)−⌊ k
2⌋ak mod 2
2 )(1n−1)),
βk
n = l0(((a1a2)⌊
k
2⌋ak mod 2
1 )(1n−1)).
Number of a1 in the expression (a1a2)⌊
k
2⌋ak mod 2
1 module 2 is equal to
the number βk
1
βk
1 =
(⌊
k
2
⌋
+ k
)
mod 2.
As a1 act not trivially on the first level only then we can omit last a1 for
n > 2
βk
n = l0(((a1a2)⌊
k
2⌋ak mod 2
1 )(1n−1)) = l0(((a1a2)⌊
k
2⌋)(1n−1)) = β
⌊ k
2⌋
n−1 .
Then we have
βk
n = β
⌊ k
2⌋
n−1 = . . . = β
⌊
k
2n−1
⌋
1 =
(⌊
k
2n
⌋
+
⌊
k
2n−1
⌋)
mod 2 for n > 1.
Now we compute αk
n for all integer k and all n > 2
αk
n = l0(((a1a2)−⌊ k
2⌋ak mod 2
2 )(1n−1)) = l0(((a1a2)−⌊ k
2⌋−k mod 2)(1n−1))
Note that
⌊
k
2
⌋
+ k mod 2 =
⌈
k
2
⌉
. Then
αk
n = β
−⌈ k
2⌉
n−1 =
(
−
⌈
k
2n
⌉
−
⌈
k
2n−1
⌉)
mod 2 =
(⌈
k
2n
⌉
+
⌈
k
2n−1
⌉)
mod 2.
Now congruence 4 looks like the following
⌈
k
2n
⌉
+
⌈
k
2n−1
⌉
+
⌊
k
2n
⌋
+
⌊
k
2n−1
⌋
+
∏
v∈T n−1
l0(((a1a2)k)(v)) ≡ 0 mod 2.
I . Samoilovych 303
We have by the properties of floor and ceil that
⌈
k
2n
⌉
−
⌊
k
2n
⌋
=
{
0, if 2n | k,
1, otherwise.
⌈
k
2n−1
⌉
−
⌊
k
2n−1
⌋
=
{
0, if 2n−1 | k,
1, otherwise.
If 2n | k then 2n−1 | k and by the Lemma 9 congruence 4 looks like the
following
⌈
k
2n
⌉
+
⌊
k
2n
⌋
+
⌈
k
2n−1
⌉
+
⌊
k
2n−1
⌋
≡ 0 + 0 ≡ 0 mod 2.
If 2n ∤ k and 2n−1 ∤ k then by the Lemma 9 congruence 4 looks like the
following
⌈
k
2n
⌉
+
⌊
k
2n
⌋
+
⌈
k
2n−1
⌉
+
⌊
k
2n−1
⌋
≡ 1 + 1 ≡ 0 mod 2.
If 2n ∤ k and 2n−1 | k then by the Lemma 9 congruence 4 looks like the
following
⌈
k
2n
⌉
+
⌊
k
2n
⌋
+
⌈
k
2n−1
⌉
+
⌊
k
2n−1
⌋
+ 1 ≡ 1 + 0 + 1 ≡ 0 mod 2.
Theorem 5. We suppose that element g = (g0, g1)σi∅ ∈ G|T [n] and
g′ = (g′
0, g
′
1)σi∅ ∈ AutT [n+1] where g′
0, g
′
1 ∈ G|T [n], g′
0|T [n−1] = g0 and
g′
1|T [n−1] = g1 for integer n > 3. Then g′ ∈ G|T [n+1] if and only if the
following congruence holds
l0(g′
(01n−1)) + l0(g′
(1n)) +
∏
v∈T n−1
l0(g(v)) ≡ 0 mod 2. (5)
Proof. Necessary condition follow from the Lemma 10.
Let us prove the sufficient condition. We know (see [7]) that |G|T [n−1] | =
2n for n > 3 and it imply that |(St(n − 1))n| = 2 for n > 3. So there
are 4 possible ways how we can construct g′ by given g. Group G
is level transitive and it imply that there are such elements h0, h1 in
St(n− 1)|T [n] that l0((h0)(1n−1)) = 0 and l0((h1)(1n−1)) = 1. It imply
that all possibilities for the pair (l0(g′
(01n−1)), l0(g′
(1n))) are realized and
304 Profinite closures of monodromy groups
pairs (l0(g′
(01n−1)), l0(g′
(1n))) are in one-to-one correspondence to g′ con-
structed in such way. For the fixed g the product
∏
v∈T n−1 l0(g(v)) is also
fixed. Then there are only two pairs (l0(g′
(01n−1)), l0(g′
(1n))) that satisfy
congruence 5. As |St(n)|T [n+1] | = 2 then there are two possibilities for
the element g′ that belong to G|T [n+1] . So element g′ that corresponds
to the pair (l0(g′
(01n−1)), l0(g′
(1n))) which satisfy congruence 5 belongs to
G|T [n+1] .
Note that group G|T [2] is AutT [2]. For all elements g ∈ G|T [3] one can
directly check following congruences
l0(g(0)) ≡ l0(g(10)) + l0(g(11)) mod 2,
l0(g(1)) ≡ l0(g(00)) + l0(g(01)) mod 2.
This congruences together with Lemma 10 and Theorem 5 give us full
explicit description of groups G|T [n] for n > 3.
References
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[4] I.V. Bondarenko, I.O. Samoilovych, On finite generation of self-similar groups
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Contact information
I. Samoilovych Department of Mechanics and Mathematics,
Kyiv National Taras Shevchenko University,
Volodymyrska, 64, Kyiv 01033, Ukraine
E-Mail(s): samoil449@gmail.com
Received by the editors: 05.04.2017.
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