On the space of subgroups of Baumslag-Solitar groups I perfect kernel and phenotype Alessandro Carderi Damien Gaboriau

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On the space of subgroups of Baumslag-Solitar

groups I: perfect kernel and phenotype

Alessandro Carderi, Damien Gaboriau,

François Le Maître and Yves Stalder

Abstract

Given a Baumslag-Solitar group, we study its space of subgroups

from a topological and dynamical perspective. We ﬁrst determine its

perfect kernel (the largest closed subset without isolated points). We

then bring to light a natural partition of the space of subgroups into

one closed subset and countably many open subsets that are invariant

under the action by conjugation. One of our main results is that the

restriction of the action to each piece is topologically transitive. This

partition is described by an arithmetically deﬁned function, that we

call the phenotype, with values in the positive integers or inﬁnity. We

eventually study the closure of each open piece and also the closure of

their union. We moreover identify in each phenotype a (the) maximal

compact invariant subspace.

Keywords: Baumslag-Solitar groups; space of subgroups; perfect kernel; topolog-

ically transitive actions; Bass-Serre theory.

MSC-classiﬁcation: 20E06; 20E08; 20F65; 37B05.

Contents

1 Introduction and presentation of the results 2

1.1 The perfect kernel ......................... 3

1.2 Dynamical partition of the perfect kernel ............ 5

1.3 Approximations by subgroups of other phenotypes ....... 7

1.4 Closures of orbits in ﬁnite phenotype .............. 8

1.5 An example: the case of BSp2,3q................ 9

arXiv:2210.14990v3 [math.GR] 8 Nov 2024

1.6 Some ideas on the techniques of proofs ............. 10

1.7 Subsequent work ......................... 12

2 Preliminaries and notations 12

2.1 Graphs and Schreier graphs ................... 13

2.2 Space of subgroups ........................ 14

2.3 Bass-Serre theory ......................... 17

3 Bass-Serre graphs 20

3.1 Pre-actions ............................ 20

3.2 Bass-Serre graphs ......................... 21

3.3 pm, nq-graphs ........................... 24

3.4 Realizing pm, nq-graphs as Bass-Serre graphs .......... 26

3.5 Additional properties of pm, nq-graphs ............. 28

3.6 Bass-Serre theory for BSpm, nq.................. 29

4 Phenotype 31

4.1 Phenotypes of natural numbers ................. 31

4.2 Phenotypes of pm, nq-graphs ................... 33

4.3 Phenotypes of BSpm, nq-actions ................. 34

4.4 Merging pre-actions ........................ 35

5 Perfect kernel and dense orbits 40

5.1 Perfect kernels of Baumslag-Solitar groups ........... 40

5.2 Phenotypical decomposition of the perfect kernel ........ 43

5.3 Closed invariant subsets with a ﬁxed ﬁnite phenotype ..... 46

6 Limits of ﬁnite phenotype subgroups 52

6.1 Limits of subgroups with ﬁxed ﬁnite phenotype ........ 52

6.2 Limits of subgroups with varying ﬁnite phenotype ....... 55

1 Introduction and presentation of the results

The Baumslag-Solitar group of non-zero integer parameters mand nis de-

ﬁned by the presentation

BSpm, nq:“@b, t|tbmt´1“bnD.(1.1)

These one-relator two-generators groups were deﬁned by Baumslag and Soli-

tar [BS62] to provide examples of groups with surprising properties, depend-

ing on the arithmetic properties of the parameters.

It results from the work of Baumslag and Solitar and of Meskin [Mes72]

that the group BSpm, nqis

•residually ﬁnite if and only if |m|“1or |n|“1or |m|“|n|;

•Hopﬁan if and only if it is residually ﬁnite or mand nhave the same

set of prime divisors.

The group BSpm, nqis amenable if and only if |m|“1or |n|“1, and

in this case, it is metabelian. All Baumslag-Solitar groups however share

weak forms of amenability: they are inner-amenable [Sta06] and a-T-menable

[GJ03].

Over the years and despite the simplicity of their presentation, these

groups have served as a standard source of examples and counter-examples,

sometimes to published results (!). They have been considered from countless

diﬀerent perspectives in group theory and beyond.

Various aspects concerning the subgroups of the BSpm, nqhave been con-

sidered such as the growth functions of their number of subgroups of ﬁnite

index with various properties, or such as a description of the kind of fun-

damental group of graphs of groups that can be embedded as subgroups in

some BSpm, nq; see for instance [Gel05,Dud09,Lev15].

In this article, we consider global aspects of the space SubpBSpm, nqq of

subgroups of the BSpm, nqand of the topological dynamics generated by the

natural action by conjugation.

1.1 The perfect kernel

Let Γbe a countable group. We denote by SubpΓqthe space of subgroups of

Γ. If one identiﬁes each subgroup with its indicator function, one can view

the space SubpΓqas a closed subset of t0,1uΓ. Thus SubpΓqis a compact,

metrizable space by giving it the restriction of the product topology. See

Section 2.2 for the generalities about SubpΓq.

By the Cantor–Bendixson theorem, SubpΓqadmits a unique decomposi-

tion as a disjoint union of a perfect set, called the perfect kernel KpΓqof

Γ, and of a countable open subset. It is a challenging problem to determine

the perfect kernel of a given countable group.

When Γis ﬁnitely generated, the ﬁnite index subgroups are isolated in

SubpΓq. It is thus relevant to introduce the subspace Subr8spΓqconsisting of

all inﬁnite index subgroups of Γ. It is a closed subspace of SubpΓqexactly

when Γis ﬁnitely generated (see Remark 2.3).

Our ﬁrst main result completely describes the perfect kernel of the various

Baumslag-Solitar groups. When |m|“|n|, the subgroup generated by bmis

normal; let us denote by πthe corresponding quotient homomorphism

BSpm, nqπ

ÑBSpm, nq{ xbmy.

We also denote by πthe map it induces between the spaces of subgroups of

BSpm, nqand BSpm, nq{ xbmy.

Theorem A (Perfect kernel of BSpm, nq, Theorem 5.3).Let m, n PZ ∖ t0u,

1. if |m|“1or |n|“1, then KpBSpm, nqq is empty;

2. if |m|,|n|ą1, then

(a) if |m|‰|n|, then KpBSpm, nqq “ Subr8spBSpm, nqq;

(b) if |m|“|n|, then KpBSpm, nqq “ π´1`Subr8spBSpm, nq{ xbmyq˘.

The fact that SubpBSpm, nqq is countable when |m|“1or |n|“1

(Item 1), i.e. for the Baumslag-Solitar groups that are metabelian, was al-

ready observed by Becker, Lubotzky, and Thom [BLT19, Corollary 8.4]. For-

tuitously or not, it turns out that KpBSpm, nqq “ Subr8spBSpm, nqq exactly

when BSpm, nqis not residually ﬁnite.

There is a general correspondence between the transitive pointed Γ-actions

and the subgroups of Γ. It sends an action αto the stabilizer of its base point.

This Γ-equivariant map is a bijection when one considers the actions up to

pointed isomorphisms (see Section 2.2). Item 2of Theorem Ahas a uniﬁed

reformulation in this setting:

2’.if |m|,|n|ą1, then KpBSpm, nqq is the space of subgroups Λsuch that

the right BSpm, nq-action on ΛzBSpm, nqhas inﬁnitely many xby-orbits.

Note that this exactly means that the quotient of the Λ-action on the stan-

dard Bass-Serre tree (see Section 2.3) of BSpm, nqis inﬁnite.

Let us now give some more context for Theorem A. By Brouwer’s char-

acterization of Cantor spaces, the space KpΓqis either empty or a Cantor

space. It is empty exactly when SubpΓqis countable. This happens for ex-

ample for groups all whose subgroups are ﬁnitely generated, also known as

Noetherian groups. For instance all ﬁnitely generated nilpotent groups and

more generally all polycyclic groups have a countable space of subgroups.

On the opposite side, for the free group with a countably inﬁnite num-

ber of generators, no subgroup is isolated, thus KpF8q “ SubpF8q(see

[CGLM23, Proposition 2.1]).

There are some classical groups for which we know that KpΓq “ Subr8spΓq.

This is the case for the free groups Fn(for 1ănă 8), see for instance

[CGLM23, Proposition 2.1]. This is also the case for the groups with inﬁnitely

many ends, for the fundamental groups of the closed surfaces of genus ě2,

and for the ﬁnitely generated LERF groups with non-zero ﬁrst ℓ2-Betti num-

ber (see [AG23]). Recall that a group Γis LERF when its set of ﬁnite index

subgroups is dense in SubpΓq(see for instance [GKM16, Theorem 3.1]).

Bowen, Grigorchuk and Kravchenko established that the perfect kernel of

the lamplighter group pZ{pZqn≀Z“ p‘ZpZ{pZqnq ¸ Z(for a prime number

p) is exactly the space Subp‘ZpZ{pZqnqof subgroups of the normal sub-

group [BGK15, Theorem 1.1]. Skipper and Wesolek uncovered the perfect

kernel for a class of branch groups containing the Grigorchuk group and the

Gupta–Sidki 3 group [SW20].

The perfect kernel can be obtained by successively, and transﬁnitely,

removing the isolated points, thus obtaining for every ordinal αthe α-th

Cantor-Bendixson derivative SubpΓqpαq:“SubpΓqpβq∖tisolated points of

SubpΓqpβquif α“β`1, and SubpΓqpαq:“ŞβăαSubpΓqpβqif αis a limit

ordinal. The Cantor–Bendixson rank rkCBpΓqof Γis the ﬁrst ordinal

ζfor which the derived space SubpΓqpζqhas no more isolated points, and

is thus equal to the perfect kernel (see for instance [Kec95, Section 6.C]

for details). When |m|,|n|ą1and |m|‰|n|, then Theorem Aimplies

that rkCBpBSpm, nqq “ 1. The determination of the Cantor-Bendixson ranks

rkCBpBSpm, nqq for the other cases is postponed to the sequel [CGLMS].

1.2 Dynamical partition of the perfect kernel

The compact space of subgroups SubpΓqis equipped with the continuous ac-

tion of Γby conjugation: Λ¨γ:“γ´1Λγ. The perfect kernel is Γ-invariant.

This action is of course not minimal in general, even when restricted to the

perfect kernel: the latter may contain normal subgroups and these subgroups

are ﬁxed points! However, the ﬁrst three named authors observed a partic-

ularly nice feature in the case of the free group Fn(for 1ănă 8): the

action KpFnqðFnis topologically transitive (which means that the space

admits a dense Gδsubset of points whose individual orbits are dense). These

Fn-actions are called totipotent, see [CGLM23].

To our surprise, we uncovered a dramatically diﬀerent and very rich sit-

uation for the Baumslag-Solitar groups.

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OnthespaceofsubgroupsofBaumslag-SolitargroupsI:perfectkernelandphenotypeAlessandroCarderi,DamienGaboriau,FrançoisLeMaîtreandYvesStalderAbstractGivenaBaumslag-Solitargroup,westudyitsspaceofsubgroupsfromatopologicalanddynamicalperspective.Wefirstdetermineitsperfectkernel(thelargestclosedsubsetwithouti...

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On the space of subgroups of Baumslag-Solitar groups I perfect kernel and phenotype Alessandro Carderi Damien Gaboriau

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