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Publicly Available Published by De Gruyter July 27, 2016

Commensurated subgroups in finitely generated branch groups

Phillip Wesolek EMAIL logo
From the journal Journal of Group Theory

Abstract

A subgroup HG is commensurated if |H:HgHg-1|< for all gG. We show that a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group and many other branch groups of independent interest is either finite or of finite index.

1 Introduction

Subgroups H and K of a group G are commensurate if |H:HK|< and |K:HK|<. The subgroup H is commensurated in G if H and gHg-1 are commensurate for all gG. Normal subgroups are obvious examples of commensurated subgroups. However, commensurated subgroups need not be even commensurate with a normal subgroup. Simple groups can admit such commensurated subgroups; for example, Thompson’s group V admits an infinite commensurated proper subgroup.

G. Margulis’ celebrated normal subgroup theorem demonstrates that any lattice in a higher rank simple algebraic group is just infinite – i.e. every non-trivial normal subgroup is of finite index; see [4]. Margulis and R. Zimmer then ask if the commensurated subgroups can be classified up to commensurability by a precise family of known commensurated subgroups; this question is sometimes called the Margulis–Zimmer commensurated subgroup problem. Aside from a strengthening of the normal subgroup theorem, the commensurated subgroup problem seems related to many aspects of arithmetic groups, as discussed in [7]. In loc. cit., Y. Shalom and G. Willis classify the commensurated subgroups for a large family of arithmetic groups, making substantial progress on this problem.

Considering the analogues of the commensurated subgroup problem for other classes of groups with few normal subgroups seems independently interesting. Using the Shalom–Willis strategy of studying certain completions, we here classify the commensurated subgroups of finitely generated just infinite branch groups. Indeed, we characterize the just infinite property for finitely generated branch groups by commensurated subgroups.

Theorem 1.1

Suppose G is a finitely generated branch group. Then G is just infinite if and only if every commensurated subgroup is either finite or of finite index.

As an immediate consequence of this result, we obtain a description of the commensurated subgroups of various groups of independent interest.

Corollary 1.2

The Grigorchuk group and the Gupta–Sidki groups are such that every commensurated subgroup is either finite or of finite index.

2 Preliminaries

We use “t.d.”, “l.c.”, and “s.c.” for “totally disconnected”, “locally compact”, and “second countable”, respectively. Recall that a t.d.l.c.s.c. group is a Polish group – i.e. it is separable and admits a complete, compatible metric.

2.1 Branch groups

Our approach to branch groups follows closely R. I. Grigorchuk’s presentation in [3].

A rooted treeT is a locally finite tree with a distinguished vertex r called the root. Letting d be the usual graph metric, the levels of T are the sets

Vn:={vTd(v,r)=n}.

The degree of a vertex vVn is the number of wVn+1 such that there is an edge from v to w. When vertices k and w lie on the same path to the root and d(k,r)d(w,r), we write kw. Given a vertex sT, the tree below s, denoted by Ts, is the collection of t such that st along with the induced graph structure.

We call a rooted tree spherically homogeneous if for all v,wVn the degree of v is the same as the degree of w. A spherically homogeneous tree is completely determined by specifying the degree of the vertices at each level. These data are given by an infinite sequence α such that α(i)2 for all i. The condition α(i)2 is to ensure that levels are not redundant; i.e. if α(i)=1, then we can remove the i-th level without changing the automorphism group. We denote a spherically homogeneous tree by Tα for α2. When αk for some k2, we write Tk.

For GAut(Tα) a subgroup and for a vertex vTα, the rigid stabilizer of v in G is defined to be

ristG(v):={gGg(w)=w for all wTαTαv}.

The n-th rigid level stabilizer in G is defined to be

ristG(n):=ristG(v)vVn.

It is easy to see that ristG(n)vVnristG(v).

Definition 2.1

A group G is said to be a branch group if there is a rooted tree Tα for some α2 such that the following hold:

  1. G is isomorphic to a subgroup of Aut(Tα).

  2. G acts transitively on each level of Tα.

  3. For each level n, the index |G:ristG(n)| is finite.

Let (iii) be the following condition:

  1. Every ristG(n) is infinite.

A group satisfying (i), (ii), and (iii) is called a weakly branch group. Plainly, every branch group is also weakly branch.

An infinite group is just infinite if all proper quotients are finite. Just infinite branch groups already have a characterization in terms of certain normal subgroups.

Theorem 2.2

Theorem 2.2 (Grigorchuk, [3, Theorem 4])

Suppose GAut(Tα) is a branch group. Then G is just infinite if and only if the commutator subgroup ristG(k) has finite index in ristG(k) for all levels k.

We shall need a fact implicit in the proof of [3, Theorem 4].

Proposition 2.3

Proposition 2.3 (Grigorchuk)

Suppose GAut(Tα) acts transitively on each level of Tα. If HG is non-trivial, then there is a level m such that ristG(m)H.

2.2 Completions and chief blocks

Our proof requires the Schlichting completion. This completion has appeared in various contexts in the literature. See for example [8] or consider [6] for a longer discussion. We here give a brief account.

Given a countable group G with a commensurated subgroup O, the group G acts by left multiplication on the collection of left cosets G/O. This induces a permutation representation σ:GSym(G/O) with kernel the normal core of O in G. The group Sym(G/O) is a topological, indeed Polish, group under the pointwise convergence topology, and we may thus form a completion as follows:

Definition 2.4

For a countable group G with a commensurated subgroup O, the Schlichting completion of G with respect to O, denoted by G//O, is defined to be σ(G)¯. The map σ:GG//O is called the completion map.

It is easy to verify that G//O is a t.d.l.c.s.c. group. When G is finitely generated, G//O is additionally compactly generated.

We shall also need the theory of chief blocks developed in [5]. A normal factor of a topological group G is a quotient K/L such that K and L are closed normal subgroups of G with L<K. We say that K/L is a chief factor if there are no closed normal subgroups of G strictly between L and K. The centralizer of a normal factor K/L is

CG(K/L):={gG[g,K]L}.

Centralizers give a notion of equivalence for chief factors; we restrict this equivalence to non-abelian chief factors for technical reasons. Non-abelian chief factors K1/L1 and K2/L2 are associated if CG(K1/L1)=CG(K2/L2). For a non-abelian chief factor K/L, the equivalence class of non-abelian chief factors equivalent to K/L is denoted by [K/L]. The class [K/L] is called a chief block of G. The set of chief blocks of G is denoted by 𝔅G. For a chief block 𝔞, the centralizer CG(𝔞) is defined to be CG(K/L) for some (equivalently, any) representative K/L.

A key property of chief blocks is a general refinement theorem.

Theorem 2.5

Theorem 2.5 (Reid–Wesolek, [5])

Let G be a Polish group, let aBG, and let {1}=G0G1Gn=G be a series of closed normal subgroups in G. Then there is exactly one i{0,,n-1} such that there exist closed normal subgroups GiB<AGi+1 of G for which A/Ba.

3 Commensurated subgroups

We first establish the reverse implication of our main theorem. For this implication, we need not assume the group is finitely generated, and the result holds for weakly branch groups.

Proposition 3.1

Let GAut(Tα) be a weakly branch group. If every commensurated subgroup of G is either finite or of finite index, then G is just infinite.

Proof.

Fix a level m. The commutator subgroup H:=ristG(m) is then a normal subgroup of G and, a fortiori, commensurated. Suppose for contradiction that H is finite. Since nmristG(n)={1}, there is some integer km such that ristG(k)H={1}. The group ristG(k) then injects into ristG(m)/H, so it is abelian. This is absurd since weakly branch groups do not admit abelian rigid stabilizers. We thus deduce that H is of finite index in G, and it follows that G is a branch group. Appealing to Theorem 2.2, G is just infinite. ∎

We now consider the converse for finitely generated branch groups.

Theorem 3.2

Suppose GAut(Tα) is a finitely generated branch group. If G is just infinite, then every commensurated subgroup of G is either finite or of finite index.

Proof.

Suppose for contradiction OG is an infinite commensurated subgroup of infinite index. Form the Schlichting completion H:=G//O and let σ:GH be the completion map.

Since G is finitely generated, H is a compactly generated t.d.l.c.s.c. group. For all open normal subgroups LH, the preimage σ-1(L) is a non-trivial normal subgroup of G, hence it has finite index. It follows that L is a finite index open subgroup of H. Every open normal subgroup of H therefore has finite index. Appealing to [1, Theorem F], we deduce that

R:={OHO is open}

is a cocompact characteristic subgroup of H without non-trivial discrete quotients. If R is trivial, then H is a compact group, and O has finite index in G. However, this is absurd, as we assume O has infinite index.

The group R is thus an infinite compactly generated t.d.l.c.s.c. group with no non-trivial discrete quotients. Since R is t.d.l.c., any non-trivial compact quotient is profinite, and thus, such a quotient produces a non-trivial discrete quotient. We deduce that R additionally has no non-trivial compact quotient. The result [1, Theorem A] now implies that R admits exactly n non-discrete topologically simple quotients where 0<n<; say that N1,,Nn lists the kernels of these quotients. The group H acts on {N1,,Nn} by conjugation, so there is a closed H~H with finite index such that H~ fixes each Ni. The pre-image σ-1(H~) is then a finite index normal subgroup of G. Via Proposition 2.3, there is some level m of the tree such that ristG(m)σ-1(H~), and we may assume m>n. Taking E:=σ(ristG(m))¯, we have that E is a finite index subgroup of H and that E normalizes each Ni. Each factor R/Ni is thus a chief factor of E; let 𝔞i be the chief block of E given by R/Ni.

For each vVm, the subgroup Lv:=σ(ristG(v))¯ is a non-abelian closed normal subgroup of E. Letting v1,,vk list Vm, put Ki:=Lv1Lvi¯ and observe that Ki<Ki+1. We thus obtain a normal series for E:

{1}<K1<<Kk=E.

Repeatedly applying Theorem 2.5, we may refine the series to include a representative for each 𝔞i. Since n<k, there is some Kj<Kj+1 such that the refinement puts no subgroups between Kj and Kj+1. The subgroup Lvj+1 is plainly contained in the centralizer of any 𝔞l which has a representative which appears in the refined series after Kj+1. On the other hand, since Lvj+1 centralizes Kj, the group Lvj+1 also centralizes the 𝔞l with representatives appearing in the refined series before Kj. The group Lvj+1 thus centralizes each block 𝔞1,,𝔞n, and therefore, it centralizes each factor R/Ni.

Returning to the setting of H, the subgroup K:=i=1nCH(R/Ni) is normal in H, and moreover, the previous paragraph ensures Lvj+1K. Therefore, K intersects σ(G) non-trivially, so σ-1(K) is a finite index subgroup of G. We conclude that K has finite index in H, so RK. Each R/Ni is thus abelian, which is absurd. ∎

Proof of Theorem 1.1.

Proposition 3.1 gives the reverse implication. Theorem 3.2 gives the forward implication. ∎

The following example shows that the finite generation hypothesis is necessary in Theorem 3.2:

Example 3.3

Let A5 be the alternating group on five elements and take the usual permutation representation (A5,[5]). For each n1, let Kn be the iterated wreath product of n copies of (A5,[5]). The permutation group given by the imprimitive action (Kn,[5]n) induces an embedding ϕn:KnAut(T5), where the action of Kn on T5 moves the vertices below the n-th level rigidly.

The Kn form a directed system, so we may take the direct limit G. One verifies that the maps ϕn cohere to induce a map ϕ:GAut(T5). The map ϕ moreover witnesses that G is a branch group. Applying Theorem 2.2, it follows that G is also just infinite.

However, G admits infinite commensurated subgroups of infinite index. For example, let F be a proper non-trivial subgroup of A5 and let (F,[5]) be the permutation representation induced by (A5,[5]). The iterated wreath products of copies of (F,[5]) again form a direct system. Moreover, the direct limit is an infinite commensurated subgroup of infinite index in G.

We conclude with an easy observation. The results of Shalom–Willis [7] show that various arithmetic groups, including SLn() for n3, have the following strong property, which is sufficient to ensure every commensurated subgroup is either finite or of finite index for a just infinite group. We say that KGcommensuratesHG if |H:HkHk-1|< for all kK.

Definition 3.4

A group G is said to have the outer commensurator-normalizer property if the following holds: for every group H and every homomorphism ψ:GH, if there is DH commensurated by ψ(G), then there is D~H commensurate with D and normalized by ψ(G).

Just infinite finitely generated branch groups can fail the outer commensurator-normalizer property. To see this, we recall a standard group-theoretic construction:

Definition 3.5

Suppose (Gi)i is a sequence of t.d.l.c. groups and suppose there is a distinguished compact open subgroup OiGi for each i. The local direct product of (Gi)i over (Oi)i is defined to be

{f:iGi|f(i)Gi, and f(i)Oi for all but finitely many i}

with the group topology such that iOi continuously embeds as an open subgroup. We denote the local direct product by i(Gi,Oi).

Local direct products of t.d.l.c. groups are again t.d.l.c. groups.

Proposition 3.6

The Grigorchuk group fails the outer commensurator-normalizer property.

Proof.

The Grigorchuk group G admits an action on a countable set X with a non-transfixed commensurated subset YX. That is to say, there is no YX such that G fixes Y setwise and |YΔY|<; see [2, Section 2.7].

Fix a non-trivial finite group F, let Fx list copies of F indexed by X, and for each xX, define

Ux:={Fif xY,{1}else.

We then form the local direct product xX(Fx,Ux). The group G obviously acts on xX(Fx,Ux) by shift, so we take

H:=xX(Fx,Ux)G.

Since Y is a commensurated subset of X and F is finite, it follows that H is a t.d.l.c. group.

The subgroup U:=xYFx is a compact open subgroup of H, and thus, G commensurates it. Suppose for contradiction that VH is commensurate with U and normalized by G. Passing to the closure if necessary, we may take V to be closed, hence V is also a compact open subgroup. Let Y be the collection of coordinates x such that the projection πx(V) is non-trivial. It follows that |YΔY|<. However, since G normalizes V, the set Y must be stabilized by G, contradicting our choice of Y. ∎


Communicated by John S. Wilson


Award Identifier / Grant number: 278469

Funding statement: The author was supported by ERC grant #278469.

Acknowledgements

The author thanks Colin Reid for his helpful remarks and the anonymous referee for his or her detailed suggestions.

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Received: 2016-2-15
Revised: 2016-6-22
Published Online: 2016-7-27
Published in Print: 2017-3-1

© 2017 by De Gruyter

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