Profinite topologies in just infinite branch groups

E.L. Pervova^{∗}

Abstract A just infinite branch group is an automorphism group of some tree. As such, it

has two natural topologies, one induced on it as a subgroup of the full automorphism group and the other its own profinite topology. A natural question is when these two topologies coincide. We answer this question positively for periodic Grigorchuk’s groups and for so-called GGS-groups and construct a class of groups for which this question is answered negatively.

Contents

1 Main definitions 3 1.1 Grigorchuk’s groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Periodic GGS-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 EGS-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Rigid stabilizers 10 2.1 Grigorchuk’s groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 GGS-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3 EGS-groups with nonsymmetric accompanying vector . . . . . . . . . . . . . 15

3 Grigorchuk’s groups 16 3.1 Groups G_{ω} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.2 The second Grigorchuk’s group. . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 GGS-groups 21 4.1 Normal subgroups of GGS-groups . . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Congruence subgroups of GGS-groups with nonsymmetric accompanying vector 22 4.3 Congruence subgroups of GGS-groups with symmetric accompanying vector 23

5 Groups without congruence property: EGS-groups 27

∗Partially supported by RFFI, grant r2001 Ural N 01-01-96404, and by Russian Ministry of Education, grant E00-1.0-150”

Introduction

A profinite group is just infinite if its every proper (continuous) quotient is finite; equivalently, if every closed normal subgroup is open. It is hereditarily just infinite if every open subgroup is just infinite.

Notice that any finitely generated profinite group which is virtually a pro-p group can be mapped onto a virtually pro-p just infinite group [6].

Another result from [6], based partly on the results of [13], states that any profinite just infinite group either contains an open normal subgroup which is isomorphic to the direct product of a finite number of copies of some hereditarily just infinite profinite group, or is a profinite branch group. The latter groups can be defined as profinite groups with a tree-like structure lattice of subnormal subgroups [14].

It is such groups that are the main subject of the present paper. More specifically, we consider abstract just infinite branch groups. As explained in some greater detail in Section 1, every such group is in fact some automorphism group G of a rooted spherically homogenous tree T (see also [1]).

As a subset of Aut T, group G is equipped with the induced topology τ_{1}. It is proven in [6] that the closure in Aut T of any branch group is a profinite branch group. However, G has its own profinite topology τ_{2} (and as G is residually finite, its completion with respect to τ_{2} is a profinite group). The main question of the current paper will be, when do the two topologies coincide?

This question admits an equivalent form. It is easy to see that the level stabilizers of vertices of T form a base of profinite topology in Aut T. Define principal congruence subgroups in G to be the intersections of those level stabilizers with G. Let us say that G has congruence property if every finite index subgroup of G is a congruence subgroup, i.e. if it contains some principal congruence subgroup. The two topologies coincide iff G has congruence property [2].

In what follows we first establish that all groups from two rather well-known classes, periodic Grigorchuk’s groups G_{ω} [5] and GGS-groups [7, 4], do possess this property. Then we introduce a class of periodic, and even p-, groups that do not have it(!). The existence of such groups is somewhat unexpected and answers negatively Question 3 of [1].

These main results are stated in Theorems 3.1, 3.2, 4.1, and 5.1. Acknowledgements A substantial part of this research has been carried out in the extremely stimulating environment of Max Planck Institute for Mathematics in Bonn whose staff I would like to thank for their kind support and hospitality. I also would like to thank A.V. Rozhkov for his attention to this work, R.I. Grigorchuk for attracting my interest to the problem, John S. Wilson, discussions with whom had helped to illuminate some tricky aspects of the subject, and Pierre de la Harpe for inviting me to University of Geneva, where this work was initiated. While at MPI, I have had a chance to explain these results to W. Gajda, A.V. Ershov, and A.Yu. Makovetskii, which eventually led to serious improvement of the expository aspect of the present paper.

1 Main definitions

Let us start with a precise definition of a branch group.

Definition 1.1. A group G is said to be a branch group, if there exist two decreasing sequences of subgroups (L_{i})_{i∈N}, (M_{i})_{i∈N} and a sequence of integers (k_{i})_{i∈N} such that

L_{0} = M_{0} = G, k_{0} = 1,
∩_{i∈N}M_{i} = 1,

and for each i the following four conditions hold,

(1) M_{i} is a normal finite-index subgroup of G;

(2) M_{i} is a direct product of k_{i} copies of the subgroup L_{i}, i.e. there are subgroups L_{i}^{(1)},

…, L_{i}^{(k_{i})} of G such that

M_{i} = L_{i}^{(1)} × … × L_{i}^{(k_{i})},  (1.1)

and each of the factors is isomorphic to L_{i};

(3) k_{i} properly divides k_{i+1}, i.e. m_{i+1} = \frac{k_{i+1}}{k_{i}} ≥ 2, and the product decomposition (1.1) of M_{i+1} refines the product decomposition (1.1) of M_{i}, meaning that each factor L_{i}^{(j)} of M_{i} contains m_{i+1} factors of M_{i+1}, namely the factors L_{i+1}^{(l)} for l = (j − 1)m_{i+1} + 1, …, jm_{i+1};

(4) conjugations by elements of G transitively permute the factors in the product decomposition (1.1).

Notice that the set of subgroups L_{i}^{(j)} ordered by inclusion forms a locally finite rooted spherically homogeneous tree T. The group G acts by automorphisms on T and is transitive on each layer of T (comprised by all subgroups L_{i}^{(j)} for a fixed i).

In general, this action may not be faithful. However, in case of just-infinite branch groups, which are what we are primarily interested in, it is faithful. Otherwise, as is easily seen, the kernel of the action would be a non-trivial normal subgroup of infinite index. Thus, a just-infinite branch group is a group of automorphisms of a rooted spherically homogeneous tree.

Let us specify what conditions a group of tree automorphisms must meet in order to be branch. Notice that if A = (A_{1}, A_{2}, …) is a sequence of finite sets of cardinality 2 or greater, then the set of all finite sequences

a_{1}a_{2} … a_{n}, a_{i} ∈ A_{i}, n ∈ N,

forms a rooted spherically homogeneous tree, where the sequences correspond to vertices, the root is the empty sequence, and two vertices are joined by an edge if they have the form a_{1}a_{2} … a_{n} and a_{1}a_{2} … a_{n}a_{n+1} for some n and some a_{i} ∈ A_{i}. For example, the tree corresponding to a branch group G can be constructed over a sequence of arbitrary sets A_{i} such that the cardinality of A_{i} is equal to m_{i}.

The number n is called the length of a vertex u = a_{1}a_{2} … a_{n} and is denoted by |u|. The set of all vertices of length n is called the nth level of T.

Of special interest is the case when all sets A_{i} are equal to one set X. In this case the tree is called regular, and the set of its vertices is precisely the set of all words in the alphabet X.

Just as we constructed a tree T over an infinite sequence of sets A = (A_{1}, A_{2}, …), we can construct a tree over a finite sequence (A_{1}, A_{2}, …, A_{n}). This tree will be denoted by T_{[n]}; geometrically, it is obtained by removing everything that lies below the nth level.

Suppose that u = â_{1}â_{2} … â_{n} is a vertex. The set of all vertices â_{1}â_{2} … â_{n}a_{n+1}a_{n+2} … a_{n+m}, where m ∈ N and a_{n+i} range over the set A_{n+i}, forms a subtree of T. We will denote that subtree by T_{u}. It is easy to see that T_{u} is naturally isomorphic to the tree T_{(n)} constructed over the sequence (A_{n+1}, A_{n+2}, …), via the map

â_{1}â_{2} … â_{n}a_{n+1}a_{n+2} … a_{n+m} ↦ a_{n+1}a_{n+2} … a_{n+m},

This map also allows to identify subtrees T_{v} for all vertices v of length n, with one fixed tree T_{(n)}.

Of course, if T is regular then for any vertex u T_{u} is isomorphic to T.

Consider now an arbitrary subgroup X in Aut T and a vertex v of T. The rigid stabilizer of v in X is the subgroup

rist_{X}(v) = {x ∈ X| for any u ∈ T \ T_{v} u^{x} = u}.

We also denote by rist_{X}(n) the subgroup ∏_{|v|=n} rist_{X}(n). This is, of course, a normal subgroup in X.

We are now ready to give an equivalent definition of a just-infinite branch group.

Definition 1.2. Let G be a group of automorphisms of a spherically homogenous tree T. G is called a branch group, if it is transitive on each level, and for every integer n ≥ 1 rist_{G}(n) has finite index in G.

There are several notions that are particularly useful in studying branch groups. Let G be a branch group and let v be a vertex of T. The stabilizer of v in G is the subgroup

st_{G}(v) = {x ∈ G|v^{x} = v}.

Denote also by st_{G}(n) the subgroup ∩_{|v|=n} st_{G}(v), which keeps all vertices of level n fixed.

Subgroups st_{G}(n) are called principal congruence subgroups in G. A subgroup of G which contains a principal congruence subgroup is called a congruence subgroup.

Definition 1.3. Group G is said to possess the congruence property if any its finite index subgroup is a congruence subgroup.

Notice that to every x ∈ st_{Aut T} (v) we can assign a unique automorphism x_{v} ∈ Aut T_{v} which is obtained by taking the restriction of x onto the subtree T_{v}. Moreover, for each vertex v of length n x_{v} can be considered as an element of Aut T_{(n)}. Thus, for each vertex v there is a fixed homomorphism φ_{v} : st_{Aut T} (v) → Aut T_{(|v|)}. We will denote φ_{v}(st_{G}(v)) by G_{v}.

In general, for different vertices u, v of the same length G_{u} and G_{v} are conjugate in Aut T_{(|u|)} but may not coincide. Let us fix for each pair of vertices u, v of the same length a conjugating element ψ_{u,v} such that G_{u}^{ψ_{u,v}} = G_{v}, and ψ_{u,u} = 1 for any u. Nonetheles, in the examples we will be considering G_{u} and G_{v} do coincide for any two vertices of the same length, and all elements ψ_{u,v} will be equal to the trivial element.

Definition 1.4. A branch group G acting on a regular tree T is fractal, if for each vertex u of T G_{u} = G.

As will be evidenced many times throughout this paper, the property of a group to be fractal is very helpful in inductive proofs.

It is clear that the direct product of φ_{v} taken over all vertices v of length n defines a monomorphism of st_{Aut T} (n) into the direct product of |A_{1}||A_{2}| … |A_{n}| copies of Aut T_{(n)}. This product map will be denoted by Φ_{n}.

Of course, collections of maps φ_{u} and Φ_{n} vary with and are determined by the tree T. Since at each given moment a tree in question will be fixed, we will keep the same notation for maps on different trees.

The above maps give a way to describe some, but not all, automorphisms of T. A more general way is given by an observation that for any vertex u of length n x induces a permutation x(u) of A_{n+1}, which acts according to the rule a ↦ ((ua)^{x})_{n+1}. Such permutations are called accompanying permutations of x. It is easy to see that they fully determine x.

Let us now describe our major examples.

1.1 Grigorchuk’s groups

We deal here with two types of examples, periodic Grigorchuk’s groups G_{ω} and the second Grigorchuk’s group G_{4}. The former are known as the first examples of groups of intermediate growth [5]. They are just-infinite branch groups ([5]) acting on the infinite binary tree. That tree can be constructed as a tree over a sequence of sets (X, X, …) for X = {0, 1}.

Groups G_{ω} are parameterized by infinite sequences ω = (ω_{1}, ω_{2}, …) such that for each i ω_{i} ∈ {0, 1, 2}, and each of the numbers 0, 1, 2 occurs in ω infinitely many times. The set of all such ω is denoted by Ω_{0}. The expression σω denotes the sequence (ω_{2}, ω_{3}, …). Clearly, σω is also in Ω_{0}.

Each Grigorchuk’s group G_{ω}, ω ∈ Ω_{0}, is generated by four automorphisms a, b_{ω}, c_{ω}, d_{ω}. (Although just three of these automorphisms are actually enough). The only nontrivial accompanying permutation of a is at the root vertex and is the cyclic permutation, denoted by π, of two elements. The other three automorphisms have infinitely many nontrivial accompanying permutations, but all of them are located in vertices u_{n} = \underbrace{11 … 1}_{n} 0, n =

0, 1, 2, . . .. Permutation b_{ω}(u_{n}) is trivial if ω_{n+1} = 2 and is equal to π otherwise. Likewise, permutation c_{ω}(u_{n}) is trivial if ω_{n+1} = 1 and is equal to π otherwise, and permutation d_{ω}(u_{n}) is trivial if ω_{n+1} = 0 and is equal to π otherwise.

Thus, for any vertex u of the form u = \underbrace{11 … 1}_{n} or u = \underbrace{11 … 1}_{n} 0 = u_{n} generators b_{ω}, c_{ω},

d_{ω} are in st_{G_{ω}}(u), and we have

φ_{u}(b_{ω}) = \{ \begin{matrix}a, & if u = u_{n} for some n and ω_{n+1} ≠ 2 \\ b_{σ_{n}ω}, & if u = \underbrace{11 … 1}_{n} \end{matrix}

φ_{u}(c_{ω}) = \{ \begin{matrix}a, & if u = u_{n} for some n and ω_{n+1} ≠ 1 \\ c_{σ_{n}ω}, & if u = \underbrace{11 … 1}_{n} \end{matrix}

φ_{u}(d_{ω}) = \{ \begin{matrix}a, & if u = u_{n} for some n and ω_{n+1} ≠ 0 \\ d_{σ_{n}ω}, & if u = \underbrace{11 … 1}_{n} \end{matrix}

In particular, for any vertex u φ_{u}(G_{ω}) = G_{σ^{n}ω} for n = |u|.

It also follows from the above definitions that exactly one of generators b_{ω}, c_{ω}, d_{ω} has trivial accompanying permutation at u_{0} = (0). That generator will be called the d-generator of G_{ω}. Now, let m(ω) be the maximal number n such that the finite sequence ω_{1}ω_{2} … ω_{n} is constant. Suppose that y_{ω} is the generator of G_{ω} such that y_{σ^{m(ω)}ω} is the d-generator of G_{σ^{m(ω)}ω}. Then y_{ω} is called the c-generator of G_{ω}. That of generators b_{ω}, c_{ω}, d_{ω} which is neither the d-generator nor the c-generator is called b-generator.

Directly applying Proposition 1 from [8], we get the following fact.

Lemma 1.1. Let G_{ω} be a periodic Grigorchuk’s group. Then

G_{ω}/[G_{ω}, G_{ω}] ≅< a > × < b_{ω}, c_{ω} >≅ Z_{2} × Z_{2} × Z_{2}.

Corollary 1.1. Let x_{ω} be the d-generator of G_{ω} and let y_{ω} be its c-generator. Then the factorgroup of G_{ω} by the normal closure N of x_{ω} is isomorphic to the group generated by a and y_{ω}.

Proof. It is sufficient to show that the intersection < a, y_{ω} > ∩N is trivial. Notice that N is contained in st_{G_{ω}}(m(ω)+1). On the other hand, the intersection < a, y_{ω} > ∩ st_{G_{ω}}(m(ω)+1) is generated by element (ay_{ω})^{2^{m(ω)+1}}. Since the order of this element is equal to 2, it is sufficient to prove that the element does not belong to N.

The image of (ay_{ω})^{2^{m(ω)+1}} under Φ_{m(ω)+1} equals (\underbrace{y_{σ^{m(ω)+1}ω}, y_{σ^{m(ω)+1}ω}, …, y_{σ^{m(ω)+1}ω}}_{2^{m(ω)+1}}). How-

ever, since x_{ω} is the d-generator of G_{ω}, for any vertex u of length m(ω) φ_{u}(N) is in the normal closure of x_{σ^{m(ω)}ω}. And since x_{σ^{m(ω)+1}ω} is not the d-generator of G_{σ^{m(ω)+1}ω}, for any vertex u of length m(ω) + 1 φ_{u}(N) is in the normal closure of the set {a, x_{σ^{m(ω)+1}ω}}. Hence the image of any φ_{u}(N) in the factorgroup of G_{σ^{m(ω)+1}ω} by its derived subgroup is in the subgroup

(< a, x_{σ^{m(ω)+1}ω} > [G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}])/[G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}].

By Lemma 1.1 we get that the image of y_{σ^{m(ω)+1}ω} is not in this subgroup, hence (ay_{ω})^{2^{m(ω)+1}} is not in N.

Let us introduce a length function ∂_{ω} on each G_{ω}. Each element of G_{ω} can be presented by a word of the form,

a^{ϵ}x_{1}a … ax_{m}a^{μ},

where x_{1}, …, x_{m} ∈ {b_{ω}, c_{ω}, d_{ω}}, and ϵ, μ are in GF(2). The length of such a word, by definition, is equal to m. The length ∂_{ω}(x) is defined as the minimum of lengths of words representing x.

The second Grigorchuk’s group G_{4} is also a 2-group, but it acts on the regular 4-tree, constructed over the sequence (X, X, …) for X = {0, 1, 2, 3}. This group is generated by two automorphisms, a and b. The former acts “at the root”, i.e. its only nontrivial accompanying permutation is located at the root vertex and equal to the cyclic permutation (0123). The second generator, b, is defined by the equality,

Φ_{1}(b) = (a, 1, a, b).

Directly applying Proposition 1 from [8], we get the following fact.

Lemma 1.2. Let G_{4} be the second Grigorchuk’s group. Then

G_{4}/[G_{4}, G_{4}] ≅< a > × < b >≅ Z_{4} × Z_{4}.

Group G_{4} is periodic [12]; using Proposition 2.2 from Section 2.1, we can prove that

Proposition 1.1. Group G_{4} is just infinite branch.

Proof. Any rist_{G_{4}}(n) has finite index in G_{4}; indeed, since G_{4} is finitely generated periodic, γ_{3}(G_{4}) has finite index in G_{4}. By Proposition 2.2 Φ_{n}(rist_{G_{4}}(n)) is equal to the direct product of 4^{n} copies of γ_{3}(G_{4}). Thus Φ_{n}(rist_{G_{4}}(n)) has finite index in Φ_{n}(st_{G_{4}}(n)), because the latter is contained in the direct product of 4^{n} copies of G_{4}. Thus, rist_{G_{4}}(n) has finite index in st_{G_{4}}(n) and hence in G_{4}. This proves that G_{4} is branch.

Again, since G_{4} is finitely generated periodic, then so is γ_{3}(G_{4}). Hence the derived subgroup [γ_{3}(G_{4}), γ_{3}(G_{4})] has finite index. Then by Theorem 4 of [6] G_{4} is just infinite.

Let us introduce a length function on G_{4}, which we will need later on. Each element x ∈ G_{4} can be represented (not uniquely) by a word of the form,

a^{δ_{1}}b^{β_{1}} … a^{δ_{m}}b^{β_{m}}a^{δ_{m+1}},

where δ_{2}, …, δ_{m}, β_{1}, …, β_{m} ∈ Z/4Z \ {0} and δ_{1}, δ_{m+1} ∈ Z/4Z. The number m will be called the length of such a word. The length ∂_{G_{4}}(x) is defined as the minimal length of words representing x.

Also, denote by ∂_{a}(x) the sum of all δ_{i} and by ∂_{b}(x) the sum of all β_{i} in some representation of x. By Lemma 1.2, the sums are independent of the choice of a particular representation, and ∂_{a}(x) and ∂_{b}(x) are well-defined.

1.2 Periodic GGS-groups

The term GGS-group was introduced by Gilbert Baumslag [4] and refers to Rostislav Grigorchuk, Narain Gupta and Said Sidki. Those groups act on regular p-trees, where p ≥ 3. We will always assume that p is prime.

Each GGS-group is fully defined by a vector

ᾱ = (α_{1}, α_{2}, …, α_{p−1}) ∈ \underbrace{GF(p) ⊕ GF(p) ⊕ … ⊕ GF(p)}_{p−1}  (1.2)

in the following way.

Construct the regular p-tree T_{p} as the tree over the sequence (X, X, …), X = {0, 1, 2, …, p − 1}. Each GGS-group Gᾱ acting on T_{p} is generated by two automorphisms a and b. The only accompanying permutation of a is at the root and is the cyclic permutation (01 … p − 1), i.e. a cyclically permutes edges coming out of the root. The second generator is in the stabilizer of level 1, and

Φ_{1}(b) = (a^{α_{1}}, a^{α^{2}}, …, a^{α_{p−1}}, b).

The vector (1.2) is called the accompanying vector of G.

By [12], all GGS-groups are just infinite branch; they are periodic if and only if ∑_{i=1}^{p−1} α_{i} = 0. Notice that for any vertex u φ_{u}(Gᾱ) = Gᾱ. Hence all Gᾱ are fractal as well.

All GGS-groups are just infinite as well; they are periodic if and only if ∑_{i=1}^{p−1} α_{i} = 0 [12]. By Proposition 1 from [8], we have

Lemma 1.3. Let Gᾱ be a GGS-group. Then

Gᾱ/[Gᾱ, Gᾱ] ≅< a > × < b >≅ Z_{p} × Z_{p}.

Later on we will need the following length function on Gᾱ. Each element x ∈ Gᾱ can be represented (not uniquely) by a word of the form,

a^{δ_{1}}b^{β_{1}} … a^{δ_{m}}b^{β_{m}}a^{δ_{m+1}},

where δ_{2}, …, δ_{m}, β_{1}, …, β_{m} ∈ GF(p) \ {0} and δ_{1}, δ_{m+1} ∈ GF(p). The number m will be called the length of such a word. The length ∂_{Gᾱ}(x) is defined as the minimal length of words representing x.

Also, denote by ∂_{a}(x) the sum of all δ_{i} and by ∂_{b}(x) the sum of all β_{i} in some representation of x. By Lemma 1.3, the sums are independent of the choice of a particular representation, and ∂_{a}(x) and ∂_{b}(x) are well-defined.

1.3 EGS-groups

The term EGS-group stands for “extended Gupta-Sidki group”, but the word “extended” should not be understood in the usual algebraic sense. While each EGS-group contains some specific GGS-group as a subgroup, it is not an extension of it. However, in a certain sense a periodic GGS-group, or, more precisely, its accompanying vector, does define a unique EGS-group.

Let Gᾱ be a periodic GGS-group with generators a and b and accompanying vector (α_{1}, α_{2}, …, α_{p−1}). The corresponding EGS-group Γᾱ is generated by automorphisms a, b, and the automorphism c such that

Φ_{1}(c) = (c, a^{α_{1}}, a^{α^{2}}, …, a^{α_{p−1}}).

The group Gᾱ is then called the associated GGS-group of Γᾱ.

Let us establish a few needed properties of EGS-groups. First of all, by Theorem 2 of [9], all EGS-groups are periodic. Just as with GGS-groups, for any vertex u φ_{u}(Γᾱ) = Γᾱ.

Lemma 1.4. Let Γᾱ be a EGS-group. Then

Γᾱ/[Γᾱ, Γᾱ] ≅< a > × < b > × < c >≅ Z/pZ × Z/pZ × Z/pZ.

Proof. Consider a group H =< a, b, c|a^{p}, b^{p}, c^{p} >, the free product of three copies of Z/pZ. Each element x of H is uniquely represented in the form

a^{s_{1}}b^{β_{1}}c^{δ_{1}} … a^{s_{m}}b^{β_{m}}c^{δ_{m}},  (1.3)

where s_{i}, β_{i}, δ_{i} ∈ GF(p), and for each i = 1, 2, …, m − 1 either β_{i} or δ_{i} is not equal to 0, for each i = 1, 2, …, m − 1 either δ_{i} or s_{i+1} is not equal to 0, and for each i = 2, 3, …, m either s_{i} or β_{i} is not equal to 0. The amount of non-zero numbers among β_{1}, β_{2}, …, β_{m}, δ_{1}, δ_{2}, …, δ_{m} is called the length of x and is denoted by ∂(x). Denote the sum of all β_{i} by ∂_{b}(x) and the sum of all δ_{i} by ∂_{c}(x).

Consider the subgroup L(H) that consists of all elements x such that s_{1}+s_{2}+…+s_{m} = 0. This is clearly a normal subgroup generated by elements b, b^{a}, …, b^{a^{p−1}}, c, c^{a}, …, c^{a^{p−1}}. Moreover, each element of L(H) is uniquely represented in the form

b^{β_{1}a^{s_{1}}}c^{δ_{1}a^{t_{1}}} … b^{β_{m}a^{sm}}c^{δ_{m}a^{tm}},  (1.4)

where s_{i}, t_{i} β_{1}, δ_{m} ∈ GF(p), and β_{2}, …, β_{m}, δ_{1}, …, δ_{m−1} are in GF(p) \ {0}. Let us denote by ∂_{b,i}(x) the sum of the β_{j} such that s_{j} = i, and by ∂_{c,i}(x) the sum of the δ_{j} such that t_{j} = i.

Consider a map Φ̃ : L(H) → \underbrace{H × … × H}_{p} acting via the rule:

Φ̃(b) = (a^{α_{1}}, …, a^{α_{p−1}}, b),
Φ̃(c) = (c, a^{α_{1}}, …, a^{α_{p−1}}),

and Φ̃(b^{a^{i}}) and Φ̃(c^{a^{i}}) are obtained from, respectively, Φ̃(b^{a^{i−1}}) and Φ̃(c^{a^{i−1}}) by one cyclic shift to the right.

Let us show first that

Φ̃^{−1}(\underbrace{[H, H] × … × [H, H]}_{p}) ≤ [H, H].

Consider an arbitrary element x ∈ L(H). Suppose that Φ̃(x) = (x_{1}, x_{2}, …, x_{p−1}, x_{0}). Then ∂_{b}(x_{i}) = ∂_{b,i}(x) and ∂_{c}(x_{i}) = ∂_{c,i}(x). Hence

∑_{i∈GF(p)} ∂_{b}(x_{i}) = ∂_{b}(x)

and

∑_{i∈GF(p)} ∂_{c}(x_{i}) = ∂_{c}(x).

Suppose that x is not in [H, H]. Then either ∂_{b}(x) or ∂_{c}(x) is not equal to zero. By the above equalities, not all of numbers ∂_{b}(x_{i}), ∂_{c}(x_{i}) are equal to zero. Hence Φ̃(x) is not in ([H, H]×…×[H, H]), which is equivalent to saying that Φ̃^{−1}([H, H]×…×[H, H]) ≤ [H, H].

Consider now the following sequence of subgroups. K_{0} = 1; K_{n} = Φ̃^{−1}(K_{n−1}×…×K_{n−1}), n ≥ 1. Take K = ∪_{n=0}^{∞}K_{n}. It is easy to show that K is a normal subgroup of H. It follows, for instance, from Proposition 1 of [9] that Γᾱ is isomorphic to the factor-group H/K. Hence Γᾱ/[Γᾱ, Γᾱ] ≅ H/[H, H], and we have the desired statement.

Relying on a fact to be established in Section 2.3, we obtain

Proposition 1.2. Let Γᾱ be a EGS-group with nonsymmetric accompanying vector. Then Γᾱ is just infinite branch.

Proof. Any rist_{Γᾱ}(n) has finite index in Γᾱ; indeed, since Γᾱ is finitely generated periodic, its derived subgroup is of finite index. By Proposition 2.4 Φ_{n}(rist_{Γᾱ}(n)) is equal to the direct product of p^{n} copies of [Γᾱ, Γᾱ]. Thus Φ_{n}(rist_{Γᾱ}(n)) has finite index in Φ_{n}(st_{Γᾱ}(n)), because the latter is contained in the direct product of p^{n} copies of Γᾱ. Thus, rist_{Γᾱ}(n) has finite index in st_{Γᾱ}(n) and hence in Γᾱ. This proves that Γᾱ is branch.

Again, since Γᾱ is finitely generated periodic, then so is its derived subgroup. Hence the second derived subgroup has finite index. Then by Theorem 4 of [6] Γᾱ is just infinite.

2 Rigid stabilizers

Knowing rigid stabilizers of a given branch group is particularly important, since by Theorem 4 of [6] their derived subgroups form the base of the profinite topology on the group in question. Moreover, it often happens that the base is formed by the rigid stabilizers themselves. As we will see in later sections, such is the case with all examples we consider.

2.1 Grigorchuk’s groups

We first compute the rigid stabilizers of the series of groups G_{ω}, ω ∈ Ω_{0}. Previously, they were only known for the first Grigorchuk’s group [10].

Proposition 2.1. Suppose n ≥ 1 and x_{σ^{n}ω} ∈ {b_{σ^{n}ω}, c_{σ^{n}ω}, d_{σ^{n}ω}} is such that x_{σ^{n−1}ω} is the d-generator of G_{σ^{n−1}ω}. Let N_{σ^{n}ω} denote the normal closure of (ax_{σ^{n}ω})^{2} in G_{σ^{n}ω}, and let y_{ω} be the d-generator of G_{ω}. Then for any vertex u of length n, if 1 ≤ n ≤ m(ω), we have

rist_{G_{ω}}(u)_{u} =< y_{σ^{n}ω}, N_{σ^{n}ω} >,  (2.1)

and if n > m(ω), we have

rist_{G_{ω}}(u)_{u} = N_{σ^{n}ω}.  (2.2)

Proof. First, we prove by induction on n that for any n and any vertex u of length n

rist_{G_{ω}}(u)_{u} ≥ N_{σ^{n}ω}.  (2.3)

The base of induction is n = 1. Let x_{ω} be the d-generator of G_{ω}. Consider element [y_{ω}^{a}, x_{ω}], where y_{ω} ∈ {b_{ω}, c_{ω}, d_{ω}} \ x_{ω}. We have

Φ_{1}([y_{ω}^{a}, x_{ω}]) = (1, (ax_{σω})^{2}).

Since for any vertex u of T st_{G_{ω}}(u)_{u} = G_{σ^{|u|}ω}, the base of induction is proven.

Assume that n > 1, and for all smaller numbers the statement is true; let us prove it for n. Since rist_{G_{ω}}(u) is normal in st_{G_{ω}}(u) and, as mentioned above, st_{G_{ω}}(u)_{u} = G_{σ^{|u|}ω}, it suffices to prove that N_{σ^{n−1}ω} contains subgroup Φ_{1}^{−1}(1×N_{σ^{n}ω}). Let y_{σ^{n−2}ω} be the d-generator of G_{σ^{n−2}ω}, and let z_{σ^{n−1}ω} be the d-generator of G_{σ^{n−1}ω}. Then N_{σ^{n−1}ω} is the normal closure of (ay_{σ^{n−1}ω})^{2}, and N_{σ^{n}ω} is the normal closure of (az_{σ^{n}ω})^{2}.

There are two possibilities.

1) Suppose that y_{σ^{n−1}ω} = z_{σ^{n−1}ω}. Then Φ_{1}((ay_{σ^{n−1}ω})^{2}) = (y_{σ^{n}ω}, y_{σ^{n}ω}). Consider some x_{σ^{n−1}ω} ∈ {b_{σ^{n−1}ω}, c_{σ^{n−1}ω}, d_{σ^{n−1}ω}} that is not the d-generator.

Notice, then, that

Φ_{1}([x_{σ^{n−1}ω}^{a}, (ay_{σ^{n−1}ω})^{2}]) = (1, (ay_{σ^{n}ω})^{2}).

Since commutator (ay_{σ^{n−1}ω})^{2} does belong to N_{σ^{n−1}ω} and the latter subgroup is normal in G_{σ^{n−1}ω}, we get the desired inclusion (2.3).

2) Suppose that y_{σ^{n−1}ω} ≠ z_{σ^{n−1}ω}. Then y_{σ^{n}ω} is not the d-generator, in particular, Φ_{1}((ay_{σ^{n−1}ω})^{2}) = (ay_{σ^{n}ω}, y_{σ^{n}ω}a). Then

Φ_{1}([(ay_{σ^{n−1}ω})^{2}, z_{σ^{n−1}ω}]) = (1, (az_{σ^{n}ω})^{2}).

Just as before, this gives us inclusion (2.3).

Let us now prove equalities (2.1) and (2.2). We start by showing that for vertex u = 1 of length 1 there is equality

rist_{G_{ω}}(u)_{u} =< y_{σω}, N_{σω} >,  (2.4)

where y_{ω} is the d-generator of G_{ω}.

Inclusion rist_{G_{ω}}(u)_{u} ≥< y_{σω}, N_{σω} > follows from (2.3) and the definition of d-generator. To prove its reverse, notice that subgroup Φ_{1}^{−1}(< y_{σω}, N_{σω}× < y_{σω}, N_{σω} >) of G_{ω} coincides with the normal closure N of y_{ω} in G_{ω}. By Corollary 1.1, the cosets of N are the elements of < a, z_{ω} >, where z_{ω} is the c-generator. A direct calculation shows that neither of those elements is in rist_{G_{ω}}(1). Thus, we have equality (2.4).

Notice that for any two vertices u and v, v = uv′,

rist_{G_{ω}}(v)_{v} = rist_{rist_{Gω} (u)_{u}}(v′)_{v′} = (rist_{G_{ω}}(u)_{u} ∩ rist_{G_{σ^{|u|}ω}}(v′))_{v′}.  (2.5)

Suppose 1 ≤ n ≤ m(ω). Then it is obvious that < y_{σ^{n−1}ω}, N_{σ^{n−1}ω} > contains Φ_{1}^{−1}(< y_{σ^{n}ω}, N_{σ^{n}ω} > × < y_{σ^{n}ω}, N_{σ^{n}ω} >), which implies equality (2.1).

Suppose now that n = m(ω)+1 and u is a vertex of length n. Let us show (2.2). It follows from (2.5) that rist_{G_{ω}}(u)_{u} is contained in < x_{σ^{n}ω}, N_{σ^{n}ω} >, where x_{σ^{n−1}ω} is the d-generator of G_{σ^{n−1}ω}. Yet, by definition of m(ω) x_{σ^{n−1}ω} does not equal y_{σ^{n−1}ω}. Hence (by Lemma 1.1) x_{σ^{n−1}ω} does not belong to < y_{σ^{n−1}ω}, N_{σ^{n−1}ω} >. Hence y_{σ^{n}ω} is not in rist_{G_{ω}}(u)_{u}, which implies (2.2).

Finally, notice that by Lemma 1.1 N_{σ^{n}ω} contains neither of generators b_{σ^{n}ω}, c_{σ^{n}ω}, d_{σ^{n}ω}, for any n. Hence we get (2.2) for all n > m(ω).

Let us now proceed to the second Grigorchuk’s group.

Lemma 2.1. The index in [G_{4}, G_{4}] of the third term γ_{3}(G_{4}) of the lower central series of G_{4} is equal to 4.

Proof. Since the derived subgroup of G_{4} is the normal closure of [b, a], the factor group [G_{4}, G_{4}]/γ_{3}(G_{4}) is cyclic, and we only need to know its order.

Let us show first that Φ_{1}(γ_{3}(G_{4})) ≥ γ_{3}(G_{4}) × γ_{3}(G_{4}) × γ_{3}(G_{4}) × γ_{3}(G_{4}). Indeed, since Φ_{1}([b^{a}, b]) = ([a, b], 1, 1, [b, a]), we have

Φ_{1}([[b, b^{a}], b^{a^{−1}}]) = (1, 1, 1, [[b, a], a]),  (2.6)
Φ_{1}([[b, b^{a}], b][[b, b^{a}], b^{a^{2}}]^{−1}) = (1, 1, 1, [[b, a], b]).  (2.7)

As γ_{3}(G_{4}) is in fact the normal closure in G_{4} of the set {[[b, a], b], [[b, a], a]}, equalities (2.6) and (2.7) give us the desired inclusion.

Consider now the element [b, a]^{4}. Since b^{4} = 1, we get that

1 = [b^{4}, a] = [b, a]^{4}(mod γ_{3}(G_{4})),

i.e. that [b, a]^{4} is in γ_{3}(G_{4}). Hence the index of γ_{3}(G_{4}) in [G_{4}, G_{4}] can be equal to 1, 2, or 4.

Let us show that [b, a]^{2} is not in γ_{3}(G_{4}). Denote by ε_{1} the natural projection G_{4} → G_{4}/ < a, [G_{4}, G_{4}] > and by ε_{2} the isomorphism between G_{4} → G_{4}/ < a, [G_{4}, G_{4}] > and the ring Z/4Z = {0, 1, 2, 3} such that ε_{2}(b) = 1. Now, let ε denote the composition (ε_{2} ∘ ε_{1} × ε_{2} ∘ ε_{1} × ε_{2} ∘ ε_{1} × ε_{2} ∘ ε_{1}) ∘ Φ_{1}. Then

ε([b, a]^{2}) = (2, 0, 0, 2),

ε([[b, a], a]) = (2, 1, 0, 1),
ε([[b, a], b]) = (0, 0, 0, 0).

Obviously, ε(st_{G_{4}}(1)) is a submodule of module Z/4Z ⊕ Z/4Z ⊕ Z/4Z ⊕ Z/4Z. Since γ_{3}(G_{4}) is the normal closure of elements [[b, a], a] and [[b, a], b], the image ε(γ_{3}(G_{4})) is the submodule generated by the elements

(2, 1, 0, 1),
(1, 2, 1, 0),
(0, 1, 2, 1),
(1, 0, 1, 2).

Thus, [b, a]^{2} is in γ_{3}(G_{4}) if and only if the following system of equations has a solution in Z/4Z:

( \begin{matrix}2 & 1 & 0 & 1 \\ 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 \\ 1 & 0 & 1 & 2 \end{matrix} ) ( \begin{matrix}x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix} ) = ( \begin{matrix}2 \\ 0 \\ 0 \\ 2 \end{matrix} )

Let us show that this system, in fact, does not have a solution. Indeed, applying elementary transformations to its extended matrix, we get

( \begin{matrix}2 & 1 & 0 & 1 & 2 \\ 1 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 \\ 1 & 0 & 1 & 2 & 2 \end{matrix} ) ⇒ ( \begin{matrix}1 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 \\ 2 & 1 & 0 & 1 & 2 \\ 1 & 0 & 1 & 2 & 2 \end{matrix} ) ⇒

⇒ ( \begin{matrix}1 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 \\ 0 & 1 & 2 & 1 & 2 \\ 0 & 2 & 0 & 2 & 2 \end{matrix} ) ⇒ ( \begin{matrix}1 & 2 & 1 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 & 2 \end{matrix} )

The resulting system, while not equivalent to the initial one, is still its consequence, and does not have a solution. Thus, the initial system does not have a solution either.

Thus, [b, a]^{2} is not in γ_{3}(G_{4}). Hence, the index of γ_{3}(G_{4}) in [G_{4}, G_{4}] is no less than 4. Earlier we proved that the index is not greater than 4 either. Thus, we have the statement of the lemma.

Remark 2.1. It can be easily inferred from the concluding arguments of the proof that for every element x of factorgroup [G_{4}, G_{4}]/γ_{3}(G_{4}) ε(x) is nontrivial. Remark 2.2. By (2.6) and (2.7) we have, in fact, that

Φ_{1}(γ_{4}(G_{4})) ≥ γ_{3}(G_{4}) × γ_{3}(G_{4}) × γ_{3}(G_{4}) × γ_{3}(G_{4}).  (2.8)

Indeed, the left parts of (2.6) and (2.7) are in γ_{4}(G_{4}).

Proposition 2.2. For any vertex u

rist_{G_{4}}(u)_{u} = γ_{3}(G_{4}).

Proof. The inclusion rist_{G_{4}}(u)_{u} ≥ γ_{3}(G_{4}) follows from (2.8). Let us prove the equality.

By (2.5), it is sufficient to prove the equality for a vertex u of length 1. To be definite, let us assume that u is the rightmost vertex. Denote by ε the natural projection of G_{4} to G_{4}/[G_{4}, G_{4}]. Then for any x ∈ st_{G_{4}}(1) the image (ε × ε × ε × ε) ∘ Φ_{1}(x) has the form,

(b^{α_{1}}a^{α_{2}+α_{4}}, b^{α_{2}}a^{α_{1}+α_{3}}, b^{α_{3}}a^{α_{2}+α_{4}}, b^{α_{4}}a^{α_{1}+α_{3}}),

where α_{1}, α_{2}, α_{3}, α_{4} ∈ Z/4Z. Thus, if x ∈ rist_{G_{4}}(u), we have α_{1} = α_{2} = α_{3} = 0 and α_{2} + α_{4} = 0. Hence α_{4} = 0 as well, and rist_{G_{4}}(u)_{u} ≤ [G_{4}, G_{4}].

Consider an arbitrary x in Φ_{1}^{−1}([G_{4}, G_{4}] × [G_{4}, G_{4}] × [G_{4}, G_{4}] × [G_{4}, G_{4}]) ∩ G_{4}. As any element of st_{G_{4}}(1), x can be represented by a word in a form,

b^{β_{1}a^{i_{1}}}b^{β_{2}a^{i_{2}}} … b^{β_{m}a^{im}},  (2.9)

where β_{j} ∈ Z/4Z \ {0}, i_{j} ∈ Z/4Z for j = 1, 2, …, m, and i_{j} ≠ i_{j+1} for j = 1, 2, …, m − 1. Denote by ∂_{i}(x), i ∈ Z/4Z, the sum of all β_{j} such that i_{j} = i. Since ∂_{i}(x) = ∂_{b}(φ_{i−1}(x)), the numbers ∂_{i}(x) do not depend on the choice of a particular representation (2.9).

For the same reason, all numbers ∂_{i}(x) are actually equal to 0 (recall that x is in Φ_{1}^{−1}([G_{4}, G_{4}] × [G_{4}, G_{4}] × [G_{4}, G_{4}] × [G_{4}, G_{4}]) ∩ G_{4}). Hence, x can be represented as a product of commutators [b^{a^{i}}, b^{a^{j}}] and their conjugates. This means, in particular, that Φ_{1}^{−1}([G_{4}, G_{4}] × [G_{4}, G_{4}] × [G_{4}, G_{4}] × [G_{4}, G_{4}]) ∩ G_{4} is the normal closure of the set of commutators [b^{a^{i}}, b^{a^{j}}].

By Lemma 2.1, the group ([G_{4}, G_{4}]/γ_{3}(G_{4}) × [G_{4}, G_{4}]/γ_{3}(G_{4}) × [G_{4}, G_{4}]/γ_{3}(G_{4}) × [G_{4}, G_{4}]/γ_{3}(G_{4})) can be regarded as module Z/4Z ⊕ Z/4Z ⊕ Z/4Z ⊕ Z/4Z. There is a natural map, which we’ll denote by ϵ, from Φ_{1}^{−1}([G_{4}, G_{4}]×[G_{4}, G_{4}]×[G_{4}, G_{4}]×[G_{4}, G_{4}])∩G_{4} to this module. The map is obtained as the composition of Φ_{1}, the fourth direct power of the natural projection [G_{4}, G_{4}] → [G_{4}, G_{4}]/γ_{3}(G_{4}), and the fourth direct power of the isomorphism [G_{4}, G_{4}]/γ_{3}(G_{4}) ≅ Z/4Z = {0, 1, 2, 3} taking [b, a]γ_{3}(G_{4}) to 1. The image of ϵ is denoted by L.

We have

ϵ([b, b^{a}]) = (3, 0, 0, 1),
ϵ([b, b^{a^{2}}]) = (0, 0, 0, 0),
ϵ([b, b^{a^{−1}}]) = (0, 0, 3, 1).

Thus, L is generated by all cyclic shifts of vectors (−1, 0, 0, 1), (0, 0, −1, 1). And since the sum of coordinates of either of these vectors is equal to 0, L does not contain vectors (0, 0, 0, 1) or (0, 0, 0, 2). This is equivalent to saying that G_{4} does not contain elements Φ_{1}^{−1}((1, 1, 1, [b, a])) or Φ_{1}^{−1}((1, 1, 1, [b, a]^{2})). Hence by Lemma 2.1, rist_{G_{4}}(u) = γ_{3}(G_{4}).

2.2 GGS-groups

The upcoming description of rigid stabilizers of GGS-groups is obtained as a partial case of Theorem 2.7.1 from [11].

Proposition 2.3. Let Gᾱ be a periodic GGS-group with accompanying vector ᾱ = (α_{1}, …, α_{p−1}). Then if ᾱ is symmetric, i.e. if α_{i} = α_{p−i} for all i = 1, 2, …, \frac{p−1}{2}, we have inclusion

Φ_{1}(γ_{3}(Gᾱ)) ≥ \underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p} .

In all other cases we have inclusion

Φ_{1}(γ_{3}(Gᾱ)) ≥ \underbrace{[Gᾱ, Gᾱ] × … × [Gᾱ, Gᾱ]}_{p} .

Proof. First, we need to check that ᾱ is not ε-diagonal, i.e. is not proportional to a vector of the form (ε, ε^{2}, …, ε^{p−1}), where ε is a unity root of pth degree in the field GF(p^{n}) for a natural n.

Since Gᾱ is periodic, α_{1} + … + α_{p−1} = 0. But the sum of coordinates of a vector (ε, ε^{2}, …, ε^{p−1}) equals to

ε + ε^{2} + … + ε^{p−1} = \{ \begin{matrix}ε\frac{1−ε^{p−1}}{1−ε} , & if ε ≠ 1 \\ p − 1 & if ε = 1 \end{matrix}

Thus, if ε = 1, this sum is not equal to 0. If ε ≠ 1, then clearly 1 − ε^{p−1} ≠ 0, and the sum is not equal to 0, either.

We obtain that ᾱ is not ε-diagonal. Then by Theorem 2.7.1 of [11], if ᾱ is ε-symmetric (i.e. has form (α_{1}, α_{2}, …, α_{s−1}, α_{s}, ε^{s}α_{s}, ε^{s−1}α_{s−1}, …, ε^{2}α_{2}, εα_{1}), s = \frac{p−1}{2}), then Φ(γ_{3}(Gᾱ)) contains the direct product \underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p}. Since the only unity root of pth degree

in GF(p) is 1, all ε-symmetric vectors are the vectors satisfying the condition α_{i} = α_{p−i}, i = 1, 2, …, \frac{p−1}{2}.

By the same theorem, in all other cases there is inclusion Φ_{1}(γ_{3}(Gᾱ)) ≥

\underbrace{[Gᾱ, Gᾱ] × … × [Gᾱ, Gᾱ]}_{p}.

Corollary 2.1. If the accompanying vector of a periodic GGS-group Gᾱ is symmetric, then for any vertex u rist_{Gᾱ}(u)_{u} contains γ_{3}(Gᾱ). Otherwise for any vertex u rist_{Gᾱ}(u)_{u} contains [Gᾱ, Gᾱ].

Proof. Follows immediately from Proposition 2.3.

2.3 EGS-groups with nonsymmetric accompanying vector

Proposition 2.4. Let Γᾱ be a EGS-group with nonsymmetric accompanying vector. Then for any vertex u rist_{Γᾱ}(u)_{u} contains [Γᾱ, Γᾱ].

Proof. Let Gᾱ be the associated GGS-group of Γᾱ. Then by Proposition 2.3 Φ_{1}(γ_{3}(Gᾱ)) ≥ \underbrace{[Gᾱ, Gᾱ] × … × [Gᾱ, Gᾱ]}_{p}. Now, we can prove in the same fashion that if Fᾱ is the sub-

group of Γᾱ generated by elements a and c, then Φ(γ_{3}(Fᾱ)) contains the direct product

\underbrace{γ_{3}(Fᾱ) × … × γ_{3}(Fᾱ)}_{p}.

Notice that [Γᾱ, Γᾱ] is the normal closure in Γᾱ of the following set: [a, b], [a, c], [b, c]. It follows from the above that for any vertex u of length 1 Φ_{1}(γ_{3}(Γᾱ)) contains elements ([a, b], 1, …, 1) and ([a, c], 1, …, 1). Further, Φ_{1}([b^{a}, c]) = ([b, c], 1, …, 1). Hence

Φ_{1}([Γᾱ, Γᾱ]) ≥ [Γᾱ, Γᾱ] × \underbrace{\{1\} × … × \{1\}}_{p−1},  (2.10)

i.e. rist_{Γᾱ}(u)_{u} ≥ [Γᾱ, Γᾱ] for all vertices of length 1. Taking into account (2.5) and inclusion (2.10), we get the desired inclusion for an arbitrary vertex u.

3 Grigorchuk’s groups

In the next two sections we establish the congruence property for three of our major examples, groups G_{ω}, the second Grigorchuk’s group, and periodic GGS-groups. In all three cases we follow a similar pattern, by proving first that rigid stabilizers are congruence subgroups (a necessary condition always) and then showing that in fact every normal subgroup of groups in question contains some rigid stabilizer (a necessary and by the first part sufficient condition).

3.1 Groups G_{ω}

Lemma 3.1. For each ω ∈ Ω_{0} the derived subgroup [G_{ω}, G_{ω}] is a congruence subgroup.

Proof. Since the derived subgroup of any G_{ω} has finite index and the intersection of a finite number of principal congruence subgroups is again a principal congruence subgroup, it is sufficient to show that any nontrivial coset of [G_{ω}, G_{ω}] has empty intersection with some principal congruence subgroup. Those cosets are a[G_{ω}, G_{ω}], ab_{ω}[G_{ω}, G_{ω}], ac_{ω}[G_{ω}, G_{ω}], ad_{ω}[G_{ω}, G_{ω}], b_{ω}[G_{ω}, G_{ω}], c_{ω}[G_{ω}, G_{ω}], d_{ω}[G_{ω}, G_{ω}]. Since [G_{ω}, G_{ω}] is contained in st_{G_{ω}}(1), the first four cosets clearly have empty intersection with st_{G_{ω}}(1).

For definiteness’ sake, let us assume that ω begins with 0. Then by definition Φ_{1}(b_{ω}) = (a, b_{σω}), Φ_{1}(c_{ω}) = (a, c_{σω}), Φ_{1}(d_{ω}) = (1, d_{σω}).

It is easy to see that [G_{ω}, G_{ω}] is generated by elements (ab_{ω})^{2}, (ac_{ω})^{2}, (ad_{ω})^{2}. By our assumption we have

Φ_{1}((ab_{ω})^{2}) = (b_{σω}a, ab_{σω}), Φ_{1}((ac_{ω})^{2}) = (c_{σω}a, ac_{σω}), Φ_{1}((ad_{ω})^{2}) = (d_{σω}, d_{σω}).

Thus, for any y ∈ [G_{ω}, G_{ω}] and for any vertex u of length 1 φ_{u}(y) belongs to one of the following cosets, [G_{σω}, G_{σω}], d_{σω}[G_{σω}, G_{σω}], ab_{σω}[G_{σω}, G_{σω}], ac_{σω}[G_{σω}, G_{σω}]. Moreover, if u and v are the two vertices of length 1, φ_{u}(y) and φ_{v}(y) belong to same coset. Thus, an element x_{ω}y, where x_{ω} ∈ {b_{ω}, c_{ω}, d_{ω}}, is in st_{G_{ω}}(2) only in case when x_{ω} = d_{ω} and φ_{u}(y) is either in d_{σω}[G_{σω}, G_{σω}] or in [G_{σω}, G_{σω}]. In the latter case one of φ_{u}(x_{ω}y), φ_{v}(x_{ω}y) is in d_{σω}[G_{σω}, G_{σω}] and the other is in [G_{σω}, G_{σω}].

Let us consider coset d_{σω}[G_{σω}, G_{σω}]. Suppose first that m(ω) = 1 (and thus ω_{2} ≠ ω_{1}). Then Φ_{1}(d_{σω}) = (a, d_{σ^{2}ω}), and, as above, for any z ∈ [G_{σω}, G_{σω}] there is a vertex u of length 1 such that φ_{u}(d_{σω}z) is not in st_{G_{σ^{2}ω}}(1). This would mean that our initial element x_{ω}y = d_{ω}y is not in st_{G_{ω}}(3).

Now we use an obvious induction on m(ω) to prove that there always is a vertex u (of length no greater than m(ω) + 1) such that φ_{u}(x_{ω}y) is not in st_{G_{σ^{|u|}ω}}(1), which implies that x_{ω}y is not in st_{G_{ω}}(|u| + 1).

Remark 3.1. It follows from the proof of the lemma that [G_{ω}, G_{ω}] contains st_{G_{ω}}(m(ω) + 2).

Proposition 3.1. For each ω ∈ Ω_{0} and each generator x_{ω} ∈ {b_{ω}, c_{ω}, d_{ω}} the normal closure of (x_{ω}a)^{2} is a congruence subgroup.

Proof. As in Lemma 3.1, consider cosets of the normal closure N of some (x_{ω}a)^{2}. Denote by y_{ω} the element of {b_{ω}, c_{ω}, d_{ω}} chosen in the following way. If x_{ω} is not the d-generator, then y_{ω} is the d-generator. If x_{ω} is the d-generator, then y_{ω} is the c-generator.

It is obvious that the factorgroup of G_{ω} by N is isomorphic to a factorgroup of finite group < x_{ω} > × < a, y_{ω} >. Fix the following system of representatives of cosets of N: a^{i}x_{ω}^{j}s, where i, j ∈ GF(2) and s ∈< y_{ω}, y_{ω}^{a} >.

Notice that since N ≤ st_{G_{ω}}(1), the desired fact has to be verified only for the cosets whose representatives are in st_{G_{ω}}(1). Thus, we can assume that i = 0.

There are two possibilities.

1) Let x_{ω} be the d-generator of G_{ω}, i.e. Φ_{1}(x_{ω}) = (1, x_{σω}). Consider t ∈ st_{G_{ω}}(1) \ N and write it as t = x_{ω}^{j}sr, where j, s are as above and r is an element of N. Let u, v be the two vertices of length 1, and let us consider the cosets containing φ_{u}(t) and φ_{v}(t).

Since Φ_{1}^{−1}([G_{σω}, G_{σω}] × [G_{σω}, G_{σω}]) ∩ G_{ω} ≤ [st_{G_{ω}}(1), st_{G_{ω}}(1)], for any two elements s and s′ in the same coset of [st_{G_{ω}}(1), st_{G_{ω}}(1)] φ_{u}(x_{ω}^{j}sr) and φ_{u}(x_{ω}^{j}s′r) are in the same coset of [G_{σω}, G_{σω}]. The same can be said about φ_{u}(x_{ω}^{j}sr) and φ_{u}(x_{ω}^{j}sr′) for any r, r′ in one coset of [st_{G_{ω}}(1), st_{G_{ω}}(1)]. Also, the same holds for the second vertex v.

In < y_{ω}, y_{ω}^{a} >, there are exactly four cosets of < y_{ω}, y_{ω}^{a} > ∩[st_{G_{ω}}(1), st_{G_{ω}}(1)]. Their representatives are, for instance, elements 1, y_{ω}, y_{ω}^{a}, y_{ω}y_{ω}^{a}. Similarly, there are exactly two cosets of N ∩ [st_{G_{ω}}(1), st_{G_{ω}}(1)] in N. Their representatives are 1 and (x_{ω}a)^{2}.

Thus, we have 16 possibilities for different choices of s and r. They are shown in the following table, where on the vertical are representatives x_{ω}^{j}s, on the horizontal are r, and the cells contain the representatives of the cosets containing φ_{u}(x_{ω}^{j}sr) and φ_{v}(x_{ω}^{j}sr).

1  (x_{ω}a)^{2}
1  (1,1)  (x_{σω}, x_{σω})
y_{ω}  (a, y_{σω})  (ax_{σω}, z_{σω})
y_{ω}^{a}  (y_{σω}, a)  (z_{σω}, ax_{σω})
y_{ω}y_{ω}^{a} (ay_{σω}, ay_{σω}) (az_{σω}, az_{σω})
x_{ω}  (1, x_{σω})  (x_{σω}, 1)
z_{ω}  (a, z_{σω})  (ax_{σω}, y_{σω})
x_{ω}y_{ω}^{a} (y_{σω}, ax_{σω})  (z_{σω}, a)
z_{ω}y_{ω}^{a} (ay_{σω}, az_{σω}) (az_{σω}, ay_{σω})

It follows from the table that, for a t in any coset other than a one of the form (y_{ω}y_{ω}^{a})^{2^{k}}N, either φ_{u}(t) or φ_{v}(t) is in a nontrivial coset of [G_{σω}, G_{σω}] and hence, by the remark after Lemma 3.1, is not in st_{G_{σω}}(m(σω) + 2). Thus, all cosets but the ones of the form (y_{ω}y_{ω}^{a})^{2^{k}}N, have empty intersection with st_{G_{ω}}(m(σω) + 3).

Let us now consider cosets (y_{ω}y_{ω}^{a})^{2^{k}}N. Notice that m(ω) is the smallest number such that y_{σ^{m(ω)}ω} is the d-generator of the relevant group. In particular, this implies that y_{ω}y_{ω}^{a} has order 2^{m(ω)+1}, and that for every k = 1, 2, …, m(ω) − 1 and for any vertex u of length k + 1 φ_{u}((y_{ω}y_{ω}^{a})^{2^{k}}) is either ay_{σ^{k+1}ω} or y_{σ^{k+1}ω}a, hence, is in the coset ay_{σ^{k+1}ω}[G_{σ^{k+1}ω}, G_{σ^{k+1}ω}]. Further, for any vertex u of length m(ω) + 1 φ_{u}((y_{ω}y_{ω}^{a})^{2^{k}}) = y_{σ^{m(ω)+1}ω} and therefore is in the coset y_{σ^{m(ω)+1}ω}[G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}].

Consider a vertex u of length k + 1, k = 1, 2, …, m(ω) and an element r of st_{N}(k + 1). Let us see which cosets of the derived subgroup [G_{σ^{k+1}ω}, G_{σ^{k+1}ω}] φ_{u}(r) can belong to. Since x_{σ^{k}ω} is the d-generator of G_{σ^{k}ω} for k = 1, 2, …, m(ω) − 1, for any element x of the normal closure of x_{σ^{k}ω} in G_{σ^{k}ω} φ_{0}(x) and φ_{1}(x) can be only in the cosets x_{σ^{k+1}ω}[G_{σ^{k+1}ω}, G_{σ^{k+1}ω}] and [G_{σ^{k+1}ω}, G_{σ^{k+1}ω}]. However, for k = 1, 2, …, m(ω) − 1 φ_{v}(r), |v| = k, actually are in the normal closure of x_{σ^{k}ω}.

Thus, for any k = 1, 2, …, m(ω)−1, any vertex u of length k+1, and for any r ∈ st_{N}(k+1) φ_{u}(r) is in a nontrivial coset of the relevant derived subgroup. By Remark 3.1 no such element is in the principal congruence subgroup st_{G_{σ^{k+1}ω}}(m(σ^{k+1}ω) + 2). And since the set (y_{ω}y_{ω}^{a})^{2^{k}}(N \ st_{N}(k + 1)) has empty intersection with st_{G_{ω}}(k + 1), we have that for k = 1, 2, …, m(ω) − 1 coset (y_{ω}y_{ω}^{a})^{2^{k}}N has empty intersection with st_{G_{ω}}(k + 1 + m(σ^{k+1}ω)).

Now let us consider coset (y_{ω}y_{ω}^{a})^{2^{m(ω)}}N. Since x_{σ^{m(ω)}ω} is not the d-generator of G_{σ^{m(ω)}ω}, for an x in the normal closure of x_{σ^{m(ω)}ω} in G_{σ^{m(ω)}ω}, φ_{0}(x) and φ_{1}(x) are in one of the following cosets, x_{σ^{m(ω)+1}ω}[G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}], a[G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}], ax_{σ^{m(ω)+1}ω}[G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}], and [G_{σ^{m(ω)+1}ω}, G_{σ^{m(ω)+1}ω}]. However, even in this case for any vertex u of length m(ω) + 1 and for any t in (y_{ω}y_{ω}^{a})^{2^{m(ω)}} st_{N}(1) φ_{u}(t) is in a nontrivial coset of the derived subgroup. Just as above, we obtain that coset (y_{ω}y_{ω}^{a})^{2^{m(ω)}}N has empty intersection with st_{G_{ω}}(m(ω) + 3 + m(σ^{m(ω)+1}ω)).

2) Suppose now that x_{ω} is not the d-generator. Then, by our choice, y_{ω} is the d-generator. This means that subgroup < y_{ω}, y_{ω}^{a} > consists of four elements, 1, y_{ω}, y_{ω}^{a}, y_{ω}y_{ω}^{a}. Let us write down a table similar to the previous one (here z_{ω} is the b-generator).

1  (x_{ω}a)^{2}
1 (1,1) (ax_{σω}, ax_{σω})
y_{ω}  (1, y_{σω}) (ax_{σω}, az_{σω})
y_{ω}^{a}  (y_{σω}, 1) (az_{σω}, ax_{σω})
y_{ω}y_{ω}^{a} (y_{σω}, y_{σω}) (az_{σω}, az_{σω})
x_{ω}  (a, x_{σω})  (x_{σω}, a)
z_{ω}  (a, z_{σω})  (x_{σω}, ay_{σω})
x_{ω}y_{ω}^{a} (ay_{σω}, x_{σω})  (z_{σω}, a)
z_{ω}y_{ω}^{a} (ay_{σω}, z_{σω}) (z_{σω}, ay_{σω})

As we see, for any element t of a nontrivial coset of N either φ_{0}(t) or φ_{1}(t) is in a nontrivial coset of [G_{σω}, G_{σω}]. Thus, by Remark 3.1 N contains principal congruence subgroup st_{G_{ω}}(m(σω) + 2).

Theorem 3.1. For each ω ∈ Ω_{0} G_{ω} has congruence property.

Proof. Since all G_{ω} are just infinite, we can just prove that the normal closure N(g) of any g ∈ G_{ω} is a congruence subgroup. We will prove this fact by induction on the length ∂_{ω}(g) of g, see Section 1.1 for definition of the length function.

Suppose g has length 1, i.e. has form a^{ϵ}x_{ω}a^{μ}, where x_{ω} ∈ {b_{ω}, c_{ω}, d_{ω}}, ϵ, μ are as above. Then its normal closure does contain the normal closure of (ax_{ω})^{2}, which is a congruence subgroup by Proposition 3.1. Hence N(g) is a congruence subgroup.

If g has length greater than 1, then by Lemma 8.2 of [5], there exist a natural number n and an element f of the normal closure of g such that f ∈ st_{G_{ω}}(u), u = \underbrace{11 … 1}_{n}, and

φ_{u}(f) ∈ {b_{σ^{n}ω}, c_{σ^{n}ω}, d_{σ^{n}ω}}. Fix some such f and choose m ≥ n such that ω_{m} ≠ ω_{m+1} and x_{σ^{m}ω} = φ_{v}(f), v = \underbrace{11 … 1}_{m}, is the d-generator.

Let y_{σ^{m−1}ω} be the d-generator of G_{σ^{m−1}ω}. Consider h = [Φ_{m}^{−1}(\underbrace{1, …, 1}_{2^{m}−1}, (ay_{σ^{m}ω})^{2}), f].

Since f ∈ st_{G_{ω}}(v), h belongs to rist_{G_{ω}}(v), and φ_{v}(h) = [(ay_{σ^{m}ω})^{2}, x_{σ^{m}ω}]. It follows from ω_{m} being not equal to ω_{m+1} and Proposition 2.1 that y_{σ^{m}ω} is not the d-generator of G_{σ^{m}ω}, and Φ_{1}(φ_{v}(h)) = (1, [y_{σ^{m+1}ω}a, x_{σ^{m+1}ω}]) = (1, (ax_{σ^{m+1}ω})^{2}).

Thus, the normal closure N(h) of h in G_{ω} contains Φ_{m+1}^{−1}(\underbrace{N_{σ^{m+1}ω} × … × N_{σ^{m+1}ω}}_{2^{m+1}}). By

Proposition 3.1, N_{σ^{m+1}ω} contains principal congruence subgroup st_{G_{σ^{m+1}ω}}(k) for some k. Hence N(h) contains Φ_{m+1}^{−1}(\underbrace{st_{G_{σ^{m+1}ω}}(k) × … × st_{G_{σ^{m+1}ω}}(k)}_{2^{m+1}}), i.e. contains st_{G_{ω}}(m + 1 + k).

Since N(g) ≥ N(h), the theorem is proven.

3.2 The second Grigorchuk’s group.

Lemma 3.2. The third term of the lower central series of G_{4} contains st_{G_{4}}(3).

Proof. Let us prove first that [G_{4}, G_{4}] contains st_{G_{4}}(2). Indeed, take x ∈ st_{G_{4}}(2). Then

Φ_{1}(x) = (b^{α_{1}}a^{α_{2}+α_{4}}x_{1}, b^{α_{2}}a^{α_{1}+α_{3}}x_{2}, b^{α_{3}}a^{α_{2}+α_{4}}x_{3}, b^{α_{4}}a^{α_{1}+α_{3}}x_{4}),

where x_{1}, x_{2}, x_{3}, x_{4} ∈ [G_{4}, G_{4}], α_{1}, α_{2}, α_{3}, α_{4} ∈ Z/4Z. We must have α_{1} + α_{3} = 0 and α_{2}+α_{4} = 0. Hence α_{1}+α_{2}+α_{3}+α_{4} = 0. However, since ∂_{b}(x) equals exactly α_{1}+α_{2}+α_{3}+α_{4}, we have that x ∈ [G_{4}, G_{4}]. Hence [G_{4}, G_{4}] contains st_{G_{4}}(2).

By Lemma 2.1 γ_{3}(G_{4}) has finite index in G_{4}. Thus, it is sufficient to prove now that every nontrivial coset of γ_{3}(G_{4}) has empty intersection with st_{G_{4}}(3).

Consider a coset of γ_{3}(G_{4}). Unless it is contained in [G_{4}, G_{4}], it has empty intersection with st_{G_{4}}(2) by the above. Suppose that it is contained in [G_{4}, G_{4}]. Then by Remark 2.1 for any element x of that coset there is a vertex u of length 1 such that φ_{u}(x) is not in [G_{4}, G_{4}], hence not in st_{G_{4}}(2). Thus, the coset has empty intersection with st_{G_{4}}(3), which is what we need.

Proposition 3.2. For any x ∈ G_{4} there is a vertex u such that st_{N}(u)_{u} contains G_{4}, where N is the normal closure of x in G.

Proof. It is sufficient to prove that there is a vertex u such that st_{N}(u)_{u} ∋ b. Indeed, in that case st_{N}(u)_{u} ∋ b^{a} as well, and therefore st_{N}(u3)_{u3} ∋ a, b.

Let us prove the desired fact by induction on the length ∂_{G_{4}}(x). The base of induction is ∂_{G_{4}}(x) ≤ 1. First suppose that ∂_{G_{4}}(x) = 0, i.e. x = a^{i}. It suffices to consider case x = a^{2}. But we have

Φ_{1}([a^{2}, b]) = (1, b^{−1}, 1, b),

and we can take the rightmost vertex as u.

Suppose that ∂_{G_{4}}(x) = 1, i.e. x = a^{i}b^{j}a^{k}. First assume that x ∈ st_{G_{4}}(1), i.e. i + k = 0. Then we can assume that x = b^{i}, and as before, we only need to consider x = b^{2}. We have

Φ_{1}([b^{2}, b^{a}]) = ([a^{2}, b], 1, 1, [b^{2}, a]),

and

Φ_{1}([a^{2}, b]) = (1, b^{−1}, 1, b),

just as we wanted.

Suppose that x is an arbitrary (of any length) element of G_{4} \ st_{G_{4}}(1). It has form a^{i}z for some i ∈ Z/4Z \ {0} and z ∈ st_{G_{4}}(1). Then

φ_{2}([x, b^{a^{1−i}}]) = \{ \begin{matrix}b, & if i = 2 \\ a, & otherwise. \end{matrix}

In the first case we have what we want right away; otherwise, as above, there exists a vertex u = 2v such that st_{N}(u)_{u} ∋ b.

We have completed the base of induction and even a part of the inductive step. Let n > 1 and ∂_{G_{4}}(x) = n. By the above, it is sufficient to consider the case when x ∈ st_{G_{4}}(1). However, for any vertex u of length 1 the length of φ_{u}(x) is less or equal to \frac{∂_{G_{4}}(x)+1}{2}, and since ∂_{G_{4}}(x) > 1, it is strictly less than ∂_{G_{4}}(x). Thus, we can apply the inductive assumption. This completes the proof of the proposition.

Theorem 3.2. The second Grigorchuk’s group has congruence property.

Proof. As any subgroup of finite index contains a normal subgroup, it is sufficient to prove that the normal closure of any element of G_{4} is a congruence subgroup. Consider the normal closure N of some x ∈ G_{4}. By Proposition 3.2, there is a vertex u such that st_{N}(u)_{u} ≥ G_{4}. Since by Proposition 2.2 rist_{G_{4}}(u)_{u} ≥ γ_{3}(G_{4}), then rist_{N}(u)_{u} ≥ γ_{4}(G_{4}). Since by Remark 2.2 Φ_{1}(γ_{4}(G_{4})) ≥ (γ_{3}(G_{4}) × γ_{3}(G_{4}) × γ_{3}(G_{4}) × γ_{3}(G_{4})), we have

N ≥ Φ_{|u|+1}^{−1}(\underbrace{γ_{3}(G_{4}) × … × γ_{3}(G_{4})}_{4^{|u|+1}}).

Then by Lemma 3.2 N ≥ st_{G_{4}}(|u| + 4).

4 GGS-groups

4.1 Normal subgroups of GGS-groups

As we mentioned in Section 1.2, all GGS-groups are fractal, i.e. for any vertex u st_{Gᾱ}(u)_{u} coincides with Gᾱ. Here we prove at first that, with the exception of a finite number of vertices, this holds for any normal subgroup of Gᾱ.

Lemma 4.1. Let Gᾱ be a periodic GGS-group. Then for any x ∈ Gᾱ there is a vertex u such that st_{X}(u)_{u} ∋ b, where X is the normal closure of x in Gᾱ.

Proof. Let us prove the statement by induction on length ∂_{Gᾱ}(x). The base of induction is ∂_{Gᾱ}(x) ≤ 1.

The elements of length 0 are a^{i}, i ∈ GF(p). We have

Φ_{1}([a^{i}, b]) = (a^{−α_{p−(i−1)}}, a^{−α_{p−(i−2)}}, …, a^{−α_{p−1}}, b^{−1}, a^{−α_{1}}, …, a^{−α_{p−i}})(a^{α_{1}}, a^{α_{2}}, …, a^{α_{p−1}}, b) =

(a^{α_{1}−α_{p−(i−1)}}, a^{α_{2}−α_{p−(i−2)}}, …, b^{−1}a^{α_{i}}, a^{α_{i+1}−α_{1}}, …, a^{−α_{p−i}}b).

If α_{i} = 0 or α_{p−i} = 0, then for either u = i − 1 or u = p − i − 1, respectively, we have φ_{u}(X) ∋ b. If α_{i}α_{p−i} ≠ 0, we proceed to case ∂_{Gᾱ}(x) = 1.

If ∂_{Gᾱ}(x) = 1, then x = a^{i}b^{ε}a^{j}. Conjugating by a^{i} if needed, we can assume that x = b^{ε}a^{j}. If j = 0, we have the desired fact right away. Suppose j ≠ 0. But in this case φ_{p−1}(x^{p}) =

b^{ε}. Since p is prime, this completes the base of induction.

Suppose now that x is an element of length m > 1, and for all elements of smaller length the statement is already proven. If x is in st_{Gᾱ}(1), for any vertex u of length 1 ∂_{Gᾱ}(φ_{u}(x)) ≤ \frac{∂_{Gᾱ}(x)+1}{2} < ∂_{Gᾱ}(x). Since Gᾱ is fractal, we can apply the inductive assumption to any φ_{u}(x).

Suppose that x is not in st_{Gᾱ}(1), i.e. x = ya^{i} for some y ∈ st_{Gᾱ}(1) and some i ∈ GF(p). Denote Φ_{1}(y) = (y_{0}, y_{1}, …, y_{p−1}). The we have

Φ_{1}(x^{p}) = (y_{0}y_{π^{i}(0)} … y_{π^{(p−1)i}(0)}, y_{1}y_{π^{i}(1)} … y_{π^{(p−1)i}(1)}, …, y_{p−1}y_{π^{i}(p−1)} … y_{π^{(p−1)i}(p−1)}),

where π denotes the (nontrivial) accompanying permutation of a. As for any element of st_{Gᾱ}(1), there is equality

∂_{a}(y_{0}) + … + ∂_{a}(y_{p−1}) = (α_{1} + … + α_{p−1})(∂_{b}(y_{0}) + … + ∂_{b}(y_{p−1})) = 0.

Therefore, since for any i ∈ GF(p) a^{i} is transitive on vertices of length 1, x^{p} is actually in st_{Gᾱ}(2). On the other hand, for any vertex u of length 1 ∂_{Gᾱ}(φ_{u}(x)) ≤ ∂_{Gᾱ}(x). Thus, for any vertex v of length 2 ∂_{Gᾱ}(φ_{v}(x)) ≤ \frac{∂_{Gᾱ}(x)+1}{2} < ∂_{Gᾱ}(x). Again, since Gᾱ is fractal, we can apply the inductive assumption to any φ_{v}(x).

Proposition 4.1. Let Gᾱ be a periodic GGS-group. Then for any x ∈ Gᾱ there is a vertex u such that st_{X}(u)_{u} = Gᾱ.

Proof. By Lemma 4.1, there is a vertex u such that st_{X}(u)_{u} ∋ b. Since Gᾱ is fractal and st_{X}(u) is normal in Gᾱ, we have st_{X}(u)_{u} ∋ b^{a^{i}} for any i ∈ GF(p) as well. Thus, if v = u(p−1) is a vertex adjacent to u, then st_{X}(v)_{v} ∋ b, a^{α_{1}}, a^{α_{2}}, …, a^{α_{p−1}}. Since among the numbers α_{1}, α_{2}, …, α_{p−1} there is a nontrivial one, st_{X}(v)_{v} = Gᾱ.

4.2 Congruence subgroups of GGS-groups with nonsymmetric accompanying vector

Despite the title of the section, the lemma below does apply to arbitrary GGS-groups.

Lemma 4.2. The derived subgroup of any periodic GGS-group Gᾱ contains principal congruence subgroup st_{Gᾱ}(2).

Proof. Consider an element x ∈ st_{Gᾱ}(1); denote Φ_{1}(x) = (x_{0}, x_{1}, …, x_{p−1}). Clearly, ∂_{b}(x) = ∑_{i=0}^{p−1} ∂_{b}(x_{i}), and vectors (∂_{b}(x_{0}), …, ∂_{b}(x_{p−1})) and (∂_{a}(x_{0}), …, ∂_{a}(x_{p−1})) are related by the following expression,

( \begin{matrix}0 & α_{p−1} & … & α_{1} \\ α_{1} & 0 & … & α_{2} \\ α_{2} & α_{1} & … & α_{3} \\ … & … & … & … \\ α_{p−2} & α_{p−3} & … & α_{p−1} \\ α_{p−1} & α_{p−2} & … & 0 \end{matrix} ) ( \begin{matrix}∂_{b}(x_{0}) \\ ∂_{b}(x_{1}) \\ ∂_{b}(x_{2}) \\ … \\ ∂_{b}(x_{p−2}) \\ ∂_{b}(x_{p−1}) \end{matrix} ) = ( \begin{matrix}∂_{a}(x_{0}) \\ ∂_{a}(x_{1}) \\ ∂_{a}(x_{2}) \\ … \\ ∂_{a}(x_{p−2}) \\ ∂_{a}(x_{p−1}) \end{matrix} ) .

For x to be in st_{Gᾱ}(2), it is necessary and sufficient that ∂_{a}(x_{i}) = 0 for all i = 0, 1, …, p − 1. Thus, x ∈ st_{Gᾱ}(2) if and only if vector (∂_{b}(x_{0}), …, ∂_{b}(x_{p−1})) is a solution to the homogenous system of linear equations with matrix

A = ( \begin{matrix}0 & α_{p−1} & … & α_{1} \\ α_{1} & 0 & … & α_{2} \\ α_{2} & α_{1} & … & α_{3} \\ … & … & … & … \\ α_{p−2} & α_{p−3} & … & α_{p−1} \\ α_{p−1} & α_{p−2} & … & 0 \end{matrix} )

over GF(p).

Assume that x ∈ st_{Gᾱ}(2); let us show that x ∈ [Gᾱ, Gᾱ]. In order to do that, it is sufficient to show that if a vector (β_{0}, …, β_{p−1}) is a solution to the aforementioned system, then ∑_{i=0}^{p−1} β_{i} = 0.

Notice that the rank of A is at least 1, i.e. the dimension of the solution space is no greater than p−1. Consider a vector (β_{0}, …, β_{p−1}) in the solution space. Since each row of A is obtained by a cyclic shift from any other row, the cyclic space generated by (β_{0}, …, β_{p−1}) is in the solution space as well. On the other hand, the dimension of this cyclic space is equal to p − GCD(x^{p} − 1, f(x)), where f(x) = β_{0} + β_{1}x + … + β_{p−1}x^{p−1}.

Since p is the characteristic of GF(p), x^{p} −1 = (x−1)^{p}. Suppose that ∑_{i=0}^{p−1} β_{i} ≠ 0. Then 1 is not a root of f(x), hence GCD(x^{p} −1, f(x)) = 1. Thus, the dimension of the cyclic space generated by (β_{0}, …, β_{p−1}) is p. But the dimension of the solution space, which contains the cyclic space, does not exceed p − 1. This contradiction proves that for any vector from the solution space the sum of its coordinates is equal to zero.

In particular, the sum of coordinates of (∂_{b}(x_{0}), …, ∂_{b}(x_{p−1})) is equal to zero. Hence, ∂_{b}(x) = 0, which, given that x is in st_{Gᾱ}(1), implies x ∈ [Gᾱ, Gᾱ].

Proposition 4.2. Any periodic GGS-group with a nonsymmetric accompanying vector has congruence property.

Proof. We need to prove that every finite index subgroup of each Gᾱ is a congruence subgroup. This is equivalent to proving that every normal finite index subgroup is a congruence subgroup, and since all Gᾱ are just infinite, we can prove that the normal closure X of any element x ∈ Gᾱ is a congruence subgroup.

By Proposition 4.1, there is a vertex u such that st_{X}(u)_{u} = Gᾱ. Notice that [st_{X}(u), rist_{Gᾱ}(u)] ≤ rist_{X}(u). By Corollary 2.1, rist_{Gᾱ}(u)_{u} ≥ [Gᾱ, Gᾱ]. Hence, rist_{X}(u)_{u} ≥ φ_{u}([st_{X}(u), rist_{Gᾱ}(u)]) ≥ γ_{3}(Gᾱ). By Proposition 2.3, γ_{3}(Gᾱ) ≥ Φ_{1}^{−1}(\underbrace{[Gᾱ, Gᾱ] × … × [Gᾱ, Gᾱ]}_{p}). Hence, there is inclusion

X ≥ Φ_{|u|+1}^{−1}(\underbrace{[Gᾱ, Gᾱ] × … × [Gᾱ, Gᾱ]}_{p^{|u|+1}}).

However, by Lemma 4.2 [Gᾱ, Gᾱ] contains st_{Gᾱ}(2). Therefore, X contains

Φ_{|u|+1}^{−1}(\underbrace{st_{Gᾱ}(2) × … × st_{Gᾱ}(2)}_{p^{|u|+1}}) = st_{Gᾱ}(|u| + 3). Theorem is proven.

4.3 Congruence subgroups of GGS-groups with symmetric accompanying vector

Lemma 4.3. Let Gᾱ be a periodic GGS-group with a symmetric accompanying vector. Then γ_{3}(Gᾱ) contains principal congruence subgroup st_{Gᾱ}(3).

Proof. Let us show first that |[Gᾱ, Gᾱ] : γ_{3}(Gᾱ)| = p. Since Gᾱ is generated by two elements a and b of finite order, it is clear that the factorgroup of [Gᾱ, Gᾱ] by γ_{3}(Gᾱ) is a finite cyclic group. The coset generating it is [a, b][Gᾱ, Gᾱ].

Let us prove that [a, b]^{p} is in γ_{3}(Gᾱ). Indeed,

1 = [a^{p}, b] = [a, b]^{p}(mod γ_{3}(Gᾱ).

Thus, |[Gᾱ, Gᾱ] : γ_{3}(Gᾱ)| equals either p or 1.

Let us show now that [a, b]^{p} is not in γ_{3}(Gᾱ). Consider map ε : st_{Gᾱ}(1) → \underbrace{GF(p) ⊕ … ⊕ GF(p)}_{p} acting via the rule, ε(x) = (∂_{b}(φ_{0}(x)), ∂_{b}(φ_{1}(x)), …, ∂_{b}(φ_{p−1}(x))).

Let L_{1} denote ε([Gᾱ, Gᾱ]) and L_{2} denote ε(γ_{3}(Gᾱ)). Since [Gᾱ, Gᾱ] is the normal closure of [a, b] in Gᾱ and

Φ_{1}([a, b]) = (b^{−1}a^{α_{1}}, a^{α_{2}−α_{1}}, …, a^{α_{p−1}−α_{p−2}}, a^{−α_{p−1}}b),

L_{1} is the cyclic space generated by vector (1, −1, 0, …, 0). Since γ_{3}(Gᾱ) is the normal closure of the set {[[a, b], a], [[a, b], b]} and

Φ_{1}([[a, b], a]) = (a^{−α_{1}}ba^{−α_{p−1}}b, a^{α_{1}−α_{2}}b^{−1}a^{α_{1}}, a^{α_{1}−α_{3}}, …, a^{α_{p−3}−α_{p−1}}, b^{−1}a^{α_{2}}),
Φ_{1}([[a, b], b]) = ([b^{−1}a^{α_{1}}, a^{α_{1}}], 1, …, 1, [a^{−α_{1}}b, b]),

L_{2} is the cyclic space generated by vector (−1, 2, −1, 0, …, 0).

Thus, the dimension of L_{1} equals p − deg f(x), where f(x) = GCD(x^{p} − 1, x − 1). Since p is equal to the charasteric of GF(p), x^{p}−1 = (x−1)^{p}. Thus, deg f(x) = 1 and dim L_{1} = p−1.

Similarly, the dimension of L_{2} equals p−deg g(x), where g(x) = GCD(x^{p} −1, x^{2} −2x+1). Since x^{2} −2x+1 = (x−1)^{2}, then deg g(x) = 2 and dim L_{2} = p−2. Hence |[Gᾱ, Gᾱ] : γ_{3}(Gᾱ)| is at least p. By the above, it has to be equal to p.

What we in fact have just proven is that every element of [Gᾱ, Gᾱ]\γ_{3}(Gᾱ) has a nontrivial image under ε. In other words, for every x in [Gᾱ, Gᾱ] \ γ_{3}(Gᾱ) there is a vertex u of length 1 such that φ_{u}(x) is not in [Gᾱ, Gᾱ].

Since by Lemma 4.2 [Gᾱ, Gᾱ] ≥ st_{Gᾱ}(2), any nontrivial coset of [Gᾱ, Gᾱ] has empty intersection with st_{Gᾱ}(2). Thus, any coset of γ_{3}(Gᾱ) which is not in [Gᾱ, Gᾱ] has empty intersection with st_{Gᾱ}(2), and nontrivial coset of γ_{3}(Gᾱ) which is in [Gᾱ, Gᾱ] has empty intersection with st_{Gᾱ}(3). Indeed, as has been noted above, for any element x of such coset there is a vertex u of length 1 such that φ_{u}(x) is not in [Gᾱ, Gᾱ] and hence is not in st_{Gᾱ}(2). Then x is not in st_{Gᾱ}(3). This proves the lemma.

Lemma 4.4. Let Gᾱ be a periodic GGS-group with a symmetric accompanying vector. Then

Φ_{1}(γ_{4}(Gᾱ)) ≥ \underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p} .

Proof. 1. By definition of a symmetric vector, α_{i} = α_{p−i}. Fix an i such that α_{p−i} ≠ 0 and consider [[b, b^{a^{p−i}}], bb^{−a^{p−2i}}]. We have

Φ_{1}(b^{a^{p−i}}) = (a^{α_{i+1}}, …, a^{α_{p−1}}, \underbrace{b}_{p−i}, a^{α_{1}}, …, a^{α_{i}}),

Φ_{1}([b, b^{a^{p−i}}]) = (\underbrace{1, …, 1}_{p−i−1}, [a^{α_{p−i}}, b], \underbrace{1, …, 1}_{i−1}, [b, a^{α_{i}}]),

Φ_{1}(b^{a^{p−2i}}) = (a^{α_{2i+1}}, …, a^{α_{p−1}}, \underbrace{b}_{p−2i}, a^{α_{1}}, …, \underbrace{a^{α_{i}}}_{p−i}, a^{α_{i}+1}, …, a^{α_{2i}}).

Since ᾱ is symmetric, we have

Φ_{1}(bb^{−a^{p−2i}}) = (\underbrace{∗, …, ∗}_{p−i−1}, 1, \underbrace{∗, …, ∗}_{i−1}, ba^{−α_{2i}}),

where stars stand for some elements of Gᾱ (we don’t care to know which). Thus,

Φ_{1}([[b, b^{a^{p−i}}], bb^{−a^{p−2i}}]) = (1, …, 1, [[b, a^{α_{i}}], ba^{−α_{2i}}]).

Put x = [[b, b^{a^{p−i}}], bb^{−a^{p−2i}}].

2. Suppose that there is j such that α_{j} = 0. Suppose also that there is i such that α_{i} = α_{p−i} ≠ 0 and α_{p−2i} = α_{2i} = 0. Then

Φ_{1}([[b, b^{a^{p−i}}], b^{a^{p−2i}}]) = (\underbrace{1, …, 1}_{p−i−1}, [[a^{α_{i}}, b], a^{α_{i}}], \underbrace{1, …, 1}_{i−1}, [[b, a^{α_{i}}], a^{α_{2i}}]) =

= (\underbrace{1, …, 1}_{p−i−1}, [[a^{α_{i}}, b], a^{α_{i}}], \underbrace{1, …, 1}_{i}).

We now show that such i can always be found (under our assumption that one of numbers α_{j} is equal to 0). It follows from a Fermat’s theorem that the remainders modulo p of numbers 2, 2^{2}, …, 2^{p−1} are all the numbers 1, 2, …, p − 1. Therefore, if we start with an arbitrary l such that α_{l} = α_{p−l} ≠ 0, the numbers α_{p−2l}, α_{p−2^{2}l}, …, α_{p−2^{p−1}l} are all the numbers α_{1}, α_{2}, …, α_{p−1} in a different order. Hence there is a k ∈ {1, 2, …, p − 1} such that α_{p−2^{k−1}l} ≠ 0 and α_{p−2^{k}l} = 0 = α_{2^{k}l}. Then the desired i is 2^{k−1}l, and we have

Φ_{1}([[b, b^{a^{p−i}}], b^{a^{p−2i}}]) = (\underbrace{1, …, 1}_{p−i}, [[a^{α_{p−i}}, b], a^{α_{p−i}}], \underbrace{1, …, 1}_{i−1}, [[b, a^{α_{i}}], a^{α_{2i}}]) =

((\underbrace{1, …, 1}_{p−i}, [[a^{α_{p−i}}, b], a^{α_{p−i}}], \underbrace{1, …, 1}_{i}).

Put y = [[b, b^{a^{p−i}}], b^{a^{p−2i}}] for this i. Notice that since γ_{3}(Gᾱ) coincides with the normal closure of [[b, a^{α_{i}}], b] and [[a^{α_{i}}, b], a^{α_{i}}], the normal closure of x, y coincides with

Φ_{1}^{−1}(\underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p}).

3. Suppose now that for any i α_{i} ≠ 0. Consider an arbitrary element of the form,

Φ_{1}(bb^{−a^{p−2j}}) = (\underbrace{∗, …, ∗}_{p−j−1}, 1, \underbrace{∗, …, ∗}_{j−1}, ba^{−α_{2j}}),

where stars stand for some elements of Gᾱ. Take an i < j. Then

Φ_{1}((bb^{−a^{p−2j}})^{a^{j−i}}) = (\underbrace{∗, …, ∗}_{p−i−1}, 1, \underbrace{∗, …, ∗}_{i−1}, a^{α_{p−j+i}−α_{p−j−i}}).

We show now that there are i and j such that α_{p−j+i} ≠ α_{p−j−i}. Assume the reverse and consider i = n, j = n + 1, 1 ≤ n < \frac{p−1}{2}. Then α_{p−1} = α_{p−1−2n}, i.e. α_{p−1} = α_{p−3} = … = α_{2}. Now consider i = n, j = n + 2, 1 ≤ n < \frac{p−3}{2}. Then α_{p−2} = α_{p−4} = … = α_{1}. On the other hand, ᾱ is symmetric, so we have α_{\frac{p−1}{2}} = α_{\frac{p+1}{2}} . As numbers \frac{p+1}{2} and \frac{p+1}{2} are of different parity, this means that all α_{i} are equal to each other, i.e. ᾱ is diagonal. However, it was shown in the proof of Proposition 2.3 that in such a case Gᾱ cannot be periodic.

Thus, we can take i and j such that α_{p−j+i} ≠ α_{p−j−i}. Then

Φ_{1}([[b, b^{a^{p−i}}], (bb^{−a^{p−2j}})^{a^{j−i}}]) = (1, …, 1, [[b, a^{α_{i}}], a^{α_{p−j+i}−α_{p−j−i}}]),

and we put y = [[b, b^{a^{p−i}}], (bb^{−a^{p−2j}})^{a^{j−i}}] for these i and j.

4. Thus, in all possible cases we have found two elements x, y such that their normal closure coincides with Φ_{1}^{−1}(\underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p}). It remains to notice that x, y are actually

in γ_{4}(Gᾱ).

We are ready to establish the main result of the section.

Proposition 4.3. Any periodic GGS-group with a symmetric accompanying vector has congruence property.

Proof. As in the nonsymmetric case, we need to prove that every normal finite index subgroup of a Gᾱ is a congruence subgroup. This is equivalent to proving that the normal closure X of any element x ∈ Gᾱ is a congruence subgroup.

By Proposition 4.1, there is a vertex u such that st_{X}(u)_{u} = Gᾱ. Again, [st_{X}(u), rist_{Gᾱ}(u)] ≤ rist_{X}(u). By Corollary 2.1, rist_{Gᾱ}(u)_{u} ≥ γ_{3}(Gᾱ). Hence, rist_{X}(u)_{u} ≥

φ_{u}([st_{X}(u), rist_{Gᾱ}(u)]) ≥ γ_{4}(Gᾱ). By Lemma 4.4, γ_{4}(Gᾱ) ≥ Φ_{1}^{−1}(\underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p}).

Hence, there is inclusion

X ≥ Φ_{|u|+1}^{−1}(\underbrace{γ_{3}(Gᾱ) × … × γ_{3}(Gᾱ)}_{p^{|u|+1}}).

However, by Lemma 4.3 γ_{3}(Gᾱ) contains st_{Gᾱ}(3). Therefore, X contains Φ_{|u|+1}^{−1}(\underbrace{st_{Gᾱ}(3) × … × st_{Gᾱ}(3)}_{p^{|u|+1}}) = st_{Gᾱ}(|u| + 4). The theorem is proven.

Combining the statements of Theorems 4.2 and 4.3, we get

Theorem 4.1. Any periodic GGS-group has congruence property.

5 Groups without congruence property: EGS-groups

In this section we establish the important result on existence of branch p-groups without the congruence property.

Lemma 5.1. Let Γᾱ be a EGS-group with nonsymmetric accompanying vector. Then for any natural number n there is an element t_{n} of coset b[Γᾱ, Γᾱ] such that t_{n} ≡ c(mod st_{Γᾱ}(n)).

Proof. Put t_{1} = b and t_{2} = b^{a}. Suppose n > 1 and an element t_{n} with the required property has already been defined. Notice that b^{−1}t_{n} is in [Γᾱ, Γᾱ]. Then by Proposition 2.4

automorphism x_{n+1} = Φ_{1}^{−1}((b^{−1}t_{n}, \underbrace{1, …, 1}_{p−1})) is in Γᾱ. Put t_{n+1} = b^{a}x_{n+1}.

We have

Φ_{1}(t_{n+1}) = (t_{n}, a^{α_{1}}, …, a^{α_{p−1}}).

Thus,

Φ_{1}(c^{−1}t_{n+1}) = (c^{−1}t_{n}, 1, …, 1).

Since by assumption c^{−1}t_{n} is in st_{Γᾱ}(n), c^{−1}t_{n+1} is in st_{Γᾱ}(n + 1).

Finally, it follows from the proof of Proposition 2.4 that x_{n+1} is in [Γᾱ, Γᾱ]. Therefore t_{n+1} is in b[Γᾱ, Γᾱ].

Corollary 5.1. The derived subgroup [Γᾱ, Γᾱ] is not a congruence subgroup.

Proof. It follows from Lemma 5.1 that for every n c^{−1}t_{n} is in st_{Γᾱ}(n). However, by Lemma 1.4 it is not in [Γᾱ, Γᾱ]. Therefore, [Γᾱ, Γᾱ] does not contain st_{Γᾱ}(n) for any n.

Thus, we obtain the following important, and surprising, result.

Theorem 5.1. No periodic EGS-group with nonsymmetric accompanying vector has congruence property.

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Author’s e-mail address: pervova@csu.ru