perso.uclouvain.be · AMENABLE UNIFORMLY RECURRENT SUBGROUPS AND LATTICE EMBEDDINGS ADRIEN LE...

AMENABLE UNIFORMLY RECURRENT SUBGROUPS ANDLATTICE EMBEDDINGS

ADRIEN LE BOUDEC

Abstract. We study lattice embeddings for the class of countable groups Γdefined by the property that the maximal amenable uniformly recurrent sub-group AΓ is continuous. When AΓ comes from an extremely proximal actionand the envelope of AΓ is co-amenable in Γ, we obtain restrictions on the lo-cally compact groups G that contain a copy of Γ as a lattice, notably regardingnormal subgroups of G, product decompositions of G, and more generally densemappings from G to a product of locally compact groups.

We then focus on a family of finitely generated groups acting on trees withinthis class, and show that these embed as cocompact irreducible lattices in somelocally compact wreath products. This provides examples of finitely generatedsimple groups quasi-isometric to a wreath product C ≀ F , where C is a finitegroup and F a non-abelian free group.

1. Introduction

The questions considered in this article fall into the setting of the followinggeneral problem: given a (class of) countable group Γ, study the locally compactgroups G such that Γ embeds as a lattice in G, i.e. such that Γ sits as a discretesubgroup of G and G/Γ carries a G-invariant probability measure.

Malcev showed that every finitely generated torsion free nilpotent group embedsas a cocompact lattice in a unique simply connected nilpotent Lie group [Rag72,Ch. II]. Conversely if G is a locally compact group with a finitely generated nilpo-tent lattice, then after modding out by a compact normal subgroup, the identitycomponent G0 is a Lie group of polynomial growth (these have been characterizedin [Gui73, Jen73]) and G/G0 is finitely generated and virtually nilpotent. Thisstatement is a combination of several works. First if G has a finitely generatednilpotent lattice Γ, then Γ is necessarily cocompact in G. Since Γ is virtually tor-sion free this is a classical fact when G is totally disconnected, and the generalcase can be deduced from [BQ14, Prop. 3.7] (which uses notably the solution ofHilbert’s fifth problem [MZ55]). In particular G is compactly generated with poly-nomial growth, and the statement then follows from the generalization of Gromov’spolynomial growth theorem for locally compact groups [Los87].

Beyond the nilpotent case, examples of classifications of embeddings of Γ as acocompact lattice have been obtained by Dymarz in [Dym15] for several familiesof examples of solvable groups Γ. Although not directly related to our concerns,we also mention that a certain dual problem was considered by Bader–Caprace–Gelander–Mozes in [BCGM16] for the class of amenable groups.

Date: February 13, 2018.This work was carried out when the author was F.R.S.-FNRS Postdoctoral Researcher.

1

2 ADRIEN LE BOUDEC

Outside the setting of amenable groups, Furman addressed the above problem forthe class of lattices Γ in semi-simple Lie groups in [Fur01], improving rigidity resultsof Mostow, Prasad, Margulis (see the references in [Fur01]; see also Furstenberg[Fur67b]). In [BFS15], Bader–Furman–Sauer considered a large class of countablegroups Γ defined by certain group theoretic conditions, and established, given alattice embedding of Γ in G, a general arithmeticity result in the setting where theconnected component of G is non-compact.

In this article we consider the class of groups whose Furstenberg uniformly recur-rent subgroup is continuous. In the first part of the article we address the questionto what extent the properties of the Furstenberg uniformly recurrent subgroup of acountable group Γ influence the locally compact groups into which Γ embeds as alattice. In the second part we focus on a family of finitely generated groups withinthis class, and explain how they embed as cocompact irreducible lattices in somelocally compact wreath products.

The groups under consideration. For a countable group Γ, the Chabauty spaceSub(Γ) of all subgroups of Γ is a compact space, on which Γ acts by conjugation.A uniformly recurrent subgroup (URS) of Γ is a closed minimal Γ-invariant subsetof Sub(Γ) [GW15]. Glasner and Weiss showed that every minimal action of Γ ona compact space X gives rise to a URS (see Proposition 3.2), called the stabilizerURS associated to the action. Conversely every URS arises as the stabilizer URSof a minimal action (see Matte Bon–Tsankov [MBT17], and Elek [Ele17] in the caseof finitely generated groups).

URS’s have been shown to be related to the study of ideals in reduced group C∗-algebras [KK17, Ken15] and reduced crossed products [Kaw17]. URS’s of severalclasses of groups have been studied in [LBMB16]. For certain examples of groupsΓ, rigidity results about minimal actions on compact spaces have been obtainedin [LBMB16] from a complete description of the space URS(Γ). Various resultsabout homomorphisms between topological full groups of étale groupoids, notablyobstructions involving invariants of the groupoids, have been obtained in [MB18]via URS’s considerations (more precisely via a complete description of the pointsin the Chabauty space of these groups whose orbit does not approach the trivialsubgroup). In the present article we will make use of URS’s as a tool in order tostudy lattice embeddings for a class of countable groups that we now define.

A URS is amenable if it consists of amenable subgroups. Every countable groupΓ admits a largest amenable URS AΓ (with respect to a natural partial orderon URS(Γ), see §3.1), which is the stabilizer URS associated to the action of Γon its Furstenberg boundary (see §2.2 for definitions). The URS AΓ is called theFurstenberg URS of Γ. AΓ is either a point, in which case we have AΓ = Rad(Γ),where Rad(Γ) is the amenable radical of Γ, or homeomorphic to a Cantor space.In this last case we say that AΓ is continuous. We refer to [LBMB16] for a moredetailed discussion.

Let (C) denote the class of groups Γ for which the Furstenberg URS AΓ iscontinuous. The class (C) is disjoint from all classes of groups previously mentionedin the introduction. More precisely, the class (C) is disjoint from the class ofamenable groups, the class of linear groups [BKKO14], and also from other classesof groups specifically considered in [BFS15], such as groups with non-vanishingℓ2-Betti numbers [BKKO14] or acylindrically hyperbolic groups (see [DGO11, Th.

AMENABLE URS’S AND LATTICE EMBEDDINGS 3

7.19] and [BKKO14, Th. 1.4]). The class (C) is stable under taking quotient byan amenable normal subgroup and extension by an amenable group [LBMB16,Prop. 2.20]. Also if Γ has a normal subgroup that is in (C), then Γ belongs to (C)[LBMB16, Prop. 2.24]. By a result of Breuillard–Kalantar–Kennedy–Ozawa, thecomplement of the class (C) is also stable under extensions (see [LBMB16, Prop.2.24]).

The study of this class of groups is also motivated by the work of Kalantar–Kennedy [KK17], who showed the following characterization: a countable group Γbelongs to (C) if and only if the group Γ/Rad(Γ) has a reduced C∗-algebra that isnot simple. For an introduction and the historical developments of the problem ofC∗-simplicity, we refer to the survey of de la Harpe [Har07].

Topological boundaries. We will make use of the notion of topological bound-ary in the sense of Furstenberg. These are compact spaces with a minimal andstrongly proximal group action (see §2.2 for definitions). Many different notions ofboundaries appear in the study of groups and group actions. What is now some-times called “boundary theory” is particularly well described in the introductionof [BF14]. We insist that in the present article the term boundary will always referto a topological boundary in the above sense. This notion should not be confusedwith any of the measured notions of boundaries. In particular, despite the possiblyconfusing terminology, the maximal topological boundary, called the Furstenbergboundary, is not the same notion as the measured notion of Poisson–Furstenbergboundary.

Lattices and direct products. Special attention will be given to products oflocally compact groups. The study of lattices in product groups is motivated(among other things) by its connections with the theory of lattices in semi-simpleLie groups, its rich geometric aspects, as well as the instances of groups with rareproperties appearing in this setting. We refer to the literature (see [Mar91, Wis96,BM97, BM00b, Rém99, Sha00, BM02, Rat04, MS04, BS06, CR09, CM12, Rad17])for developments over the last years on the study of lattices in products of locallycompact groups.

Given a countable group Γ with a continuous Furstenberg URS and a group Gcontaining Γ as a lattice, we are interested in understanding how close the groupG can be from a direct product of two groups, or which properties the group Gcan share with a direct product. Of course various notions of closeness can beconsidered. The most basic one is to ask whether the group G admits non-trivialdecompositions as a direct product. One step further, one might consider quotientmorphisms from G onto direct products of groups. In Theorems 1.1 and 1.2 belowwe more generally consider continuous morphisms with dense image from G to adirect product of groups G → G1 ×G2. We make no assumption about injectivityof these maps or injectivity of the composition with the projection to one factorGi. In particular this setting allows maps of the form G → G/N1 ×G/N2 for closednormal subgroups N1, N2 such that N1N2 is dense in G.

First results. A central notion in this article is the one of extremely proximalaction. Minimal and extremely proximal actions naturally arise in geometric grouptheory, and are boundaries in the sense of Furstenberg. We refer to §2.3 for defini-tions and examples. We say that the Furstenberg URS AΓ of a countable group Γ

4 ADRIEN LE BOUDEC

comes from an extremely proximal action if there exists a compact space Z and aΓ-action on Z that is minimal and extremely proximal, whose associated stabilizerURS is equal to AΓ. Note that typically Z will not be the Furstenberg boundaryof Γ. If H is a URS of Γ, the envelope Env(H) of H is by definition the subgroupof Γ generated by all the subgroups H ∈ H.Theorem 1.1. Let Γ be a countable group whose Furstenberg URS comes froma faithful and extremely proximal action, and let G be a locally compact groupcontaining Γ as a lattice. The following hold:

(a) Assume that Env(AΓ) is finitely generated and co-amenable in Γ. Then Gcannot be a direct product G = G1 ×G2 of two non-compact groups.

(b) Assume that Env(AΓ) has finite index in Γ and finite abelianization. Thenany continuous morphism with dense image from G to a product of locallycompact groups G → G1 ×G2 is such that one factor Gi is compact.

The same conclusions hold for any group commensurable to G up to compact ker-nels.

This result has applications to the setting of groups acting on trees, see Corollary1.3. We make several comments about the theorem:1) We do not assume that Γ is finitely generated, nor that G is compactly gen-

erated. For statement (a), The assumption that Env(AΓ) if finitely generatedadmits variations, see Theorem 5.14.

2) Making an assumption on the “size” of the envelope of AΓ with respect to Γ isnatural, in the sense that in general there is no hope to derive any conclusion onthe entire group Γ if this envelope is too small. An extreme illustration of thisis that there are groups Γ whose Furstenberg URS comes from a faithful andextremely proximal action but is trivial, and these can be lattices in products,e.g. PSL(2,Z[1/p]) inside PSL(2,R) × PSL(2,Qp) (see also the discussion rightafter Corollary 1.3).

3) Under the assumption that Env(AΓ) is co-amenable in Γ, the fact that theFurstenberg URS AΓ comes from a faithful and extremely proximal actionis equivalent to asking that the action of Γ on AΓ is faithful and extremelyproximal; see Remark 3.27. This provides an intrinsic reformulation of theassumption not appealing to any auxiliary space.

4) For Γ as in the theorem, the assumption in statement (b) that Env(AΓ) hasfinite index in Γ and Env(AΓ) has finite abelianization is equivalent to Γ beingvirtually simple (see Proposition 4.6).

The URS approach to study lattice embeddings allows to consider more gener-ally subgroups of finite covolume. Recall that a closed subgroup H of a locallycompact group G has finite covolume in G if G/H carries a G-invariant prob-ability measure. Thus a lattice is a discrete subgroup of finite covolume. Beforestating the following result we need some terminology.

Recall the notion of disjointness introduced by Furstenberg in [Fur67a]. If X,Yare compact G-spaces, X and Y are disjoint if whenever Ω is a compact G-spaceand Ω → X and Ω → Y are continuous equivariant surjective maps, the mapΩ → X × Y that makes the natural diagram commute remains surjective (see


§3.3). When X,Y are minimal G-spaces, this is equivalent to asking that thediagonal G-action on the product X × Y is minimal.

Consider the following property: two non-trivial G-boundaries are never dis-joint. A group with this property will be called boundary indivisible. Glasnercharacterized minimal compact G-spaces which are disjoint from all G-boundariesas those carrying a fully supported measure whose orbit closure in the space ofprobability measures is minimal [Gla75, Th. 6.2]. The relation between disjoint-ness and boundaries that we consider here is of different spirit, as it deals withdisjointness within the class of G-boundaries, rather than disjointness from thisclass. Locally compact groups with a cocompact amenable maximal subgroup areexamples of boundary indivisible groups [Fur73, Prop. 4.4]. On the contrary, mostdiscrete groups are not boundary indivisible. The relevance of this property in oursetting comes from the fact that, as we will show in Proposition 3.24, a discretegroup Γ as in Theorem 1.1 is boundary indivisible. Actually the only examples of(non-amenable) boundary indivisible discrete groups that we are aware of fall intothe setting of Proposition 3.24.

Recall that a convex compact G-space is irreducible if it does not contain anyproper closed convex G-invariant subspace. We say that a subgroup L of a topolog-ical group G is weakly co-amenable in G if whenever Q is a non-trivial convexcompact G-space in which L fixes a point, Q is not irreducible. This is indeed aweakening of the notion of co-amenability †, which asks that every convex compactG-space Q with L-fixed points has G-fixed points [Eym72] (and hence Q is notirreducible, unless trivial). If G has a subgroup that is both amenable and weaklyco-amenable, thenG is amenable; and a normal weakly co-amenable subgroup is co-amenable. However in general weak co-amenability does not imply co-amenability,even for discrete groups. In §6.3 we exhibit examples of finitely generated groupssuch that every subgroup is either amenable or weakly co-amenable, but havingnon-amenable subgroups that are not co-amenable.

Finally we say that a subgroup L ≤ G is boundary-minimal if there exists anon-trivial G-boundary on which L acts minimally. We refer to §5.1 for contextand examples.

Theorem 1.2. Let H be a locally compact group with an amenable URS that comesfrom an extremely proximal action, and whose envelope is co-amenable in H. LetG be a locally compact group containing H as a closed subgroup of finite covolume.Then G is boundary indivisible, and the following hold:

(a) Whenever G → G1 ×G2 is a continuous morphism with dense image, onefactor Gi is amenable.

(b) If L is a boundary-minimal subgroup of G, and L is uniformly recurrent,then L is weakly co-amenable in G. In particular every boundary-minimalnormal subgroup of G is co-amenable in G.

Again we make several comments:1) The group H is allowed to be discrete, so the theorem applies for all groups

Γ as in Theorem 1.1. While (a) will be an intermediate step in the proof ofTheorem 1.1, (b) provides additional information that is rather independentof the conclusion of Theorem 1.1.

†and not a relative version of a notion of weak amenability.

6 ADRIEN LE BOUDEC

2) Statement (a) implies that whenever N1, N2 are closed normal subgroups of Gsuch that N1N2 is dense in G, at least one Ni must be co-amenable in G.

3) The last sentence in (b) implies that if N is a closed normal subgroup of Gsuch that NCN is open in G (where CN is the centralizer of N), then N iseither amenable or co-amenable in G (see Proposition 5.3). We do not knowwhether the condition that NCN is open can be removed.

4) Theorem 1.2 does not say anything about amenable normal subgroups of G. Itis worst pointing out that, as illustrated by the examples discussed in Section6, it happens that a discrete group Γ satisfying the assumptions of Theorem1.2 and with trivial amenable radical, sits as a lattice in a group G with non-compact amenable radical.

5) Remark 5.13 below provides counter-examples showing that in statement (b)the conclusion cannot be strengthened by saying that L is co-amenable in G.

6) For H,G as in the theorem, it happens that G splits as a direct product of twonon-compact groups, even under the additional assumption that the amenableURS of H comes from a faithful extremely proximal action (see Example 6.3),so that “amenable” cannot be replaced by “compact” in statement (a).

We view the above remarks 4)-5)-6) as illustrations of the limitations of the useof topological boundaries and URS’s to the problem addressed here in the ratherabstract setting of Theorem 1.2.

Group actions on trees is a natural source of extremely proximal actions, andTheorems 1.1 and 1.2 find applications in this setting. In the following statementT is a locally finite simplicial tree.

Corollary 1.3. Let Γ ≤ Aut(T ) be a countable group having no proper invariantsubtree and no finite orbit in T ∪ ∂T . Assume that Γξ is non-trivial and amenablefor all ξ ∈ ∂T ; and Γ is virtually simple. If G is a locally compact group containingΓ as a lattice, then:

(a) any continuous morphism with dense image G → G1 ×G2 is such that onefactor Gi is compact. In particular G itself cannot be a direct product oftwo non-compact groups.

(b) The conclusion (b) of Theorem 1.2 holds for G.

A group Γ as in Corollary 1.3 is never discrete in Aut(T ). Recall that Burgerand Mozes constructed simple groups Γ acting on two locally finite regular treesT, T ′ such that the image of Γ in Aut(T ) and Aut(T ′) are non-discrete, but Γ actsfreely and cocompactly on T × T ′, so that Γ is a cocompact lattice in the productAut(T )×Aut(T ′) [BM00b]. These examples illustrate the fact that the assumptionin Corollary 1.3 that end-stabilizers are all non-trivial is essential.

Examples of groups to which Corollary 1.3 applies can be found among the fam-ily of groups denoted G(F, F ′) in [LB16] (see Corollary 6.22). These are examplesof groups with a continuous Furstenberg URS. Here F ′ ≤ Sym(d) is a finite permu-tation group and F is a simply transitive (or regular) subgroup of F ′. The groupG(F, F ′) is then a finitely generated group acting on a d-regular tree, transitivelyon vertices and edges, and with local action at every vertex isomorphic to F ′. We


refer to §6.2 for a definition. The normal subgroup structure of these groups ishighly sensible to the permutation groups: there are permutation groups F, F ′

such that G(F, F ′) virtually admits a non-abelian free quotient (Proposition 6.11),and there are permutation groups F, F ′ such that G(F, F ′)∗ (the subgroup of indextwo in G(F, F ′) preserving the bipartition of Td) is simple [LB16, Cor. 4.14]. Thisfamily of groups and the family of Burger–Mozes lattices both contain instancesof finitely generated simple groups which embed densely in some universal groupU(F )+ [BM00a, §3.2]. Despite these similarities, Corollary 6.22 shows that anygroup containing a virtually simple G(F, F ′) as a lattice is rather allergic to anydirect product behavior. Compare with Theorem 1.4.

We also mention that other examples of groups to which Corollary 1.3 can beapplied may be found among the family of piecewise prescribed tree automorphismgroups considered in [LB17, Sec. 4].

Irreducible lattices in wreath products. Leaving aside the previous abstractsituation, we then focus on the family of groups G(F, F ′) (see §6.2 for definitions).The above mentioned common properties between the discrete groups G(F, F ′) andcertain lattices in the product of two trees provide motivation for studying whichlocally compact groups can contain a group G(F, F ′) as a lattice. The contributionof this article to this problem is on the one hand the conclusions given by Corollary1.3 (see Corollary 6.22), and on the other hand to describe embeddings of thesegroups as irreducible lattices in some locally compact wreath products.

If H is a group acting on a set Ω, and A is a subgroup of a group B, the semi-restricted permutational wreath product B ≀AΩ H, introduced by Cornulier in[Cor17], is the semi-direct product BΩ,A o H, where BΩ,A is the set of functionsf : Ω → B such that f(x) ∈ A for all but finitely many x ∈ Ω, and H acts on BΩ,A

in the usual way. This definition somehow interpolates between the restricted andthe unrestricted permutational wreath products, which correspond respectively toA = 1 (in which case we will write B ≀Ω H) and A = B. When B,H are locallycompact and A is compact open in B, there is a natural locally compact grouptopology on B ≀AΩ H (see §6.5.1).

We call a lattice Γ in B ≀AΩH irreducible if Γ has a non-discrete projection to H.The terminology is motivated by the fact that this definition prevents Γ, and moregenerally any subgroup commensurable with Γ, from being of the form Γ1 o Γ2,where Γ1 and Γ2 are lattices in BΩ,A and H.

In the following statement Ck is the cyclic group of order k, Sk the symmetricgroup on k elements, Vd the vertex set of a d-regular tree Td, and G(F, F ′)∗ thesubgroup of index two in G(F, F ′) preserving the bipartition of Td.

Theorem 1.4. Let d ≥ 3, F F ′ ≤ Sym(d) permutation groups such that F actsfreely, and n the index of F in F ′. Then the following hold:

(a) G(F, F ′) embeds as an irreducible cocompact lattice in the semi-restrictedpermutational wreath product Gn,d = Sn ≀Sn−1

VdAut(Td).

(b) When F is a transitive permutation group, the finitely generated group

Γn,d = Cn ≀Vd(Cd ∗ Cd) = (C2

n ≀ Fd−1) o Cd

and G(F, F ′)∗ have isometric Cayley graphs.

8 ADRIEN LE BOUDEC

We note that no finite index subgroup of Gn,d can split non-trivially as a product,but the stabilizer of an edge of Td in Gn,d (for the projection action on Td) is anopen subgroup which does split as a direct product of two non-compact groups.

The embedding of G(F, F ′) in Gn,d = SVd,Sn−1n o Aut(Td) is not the inclusion

in the subgroup 1 o Aut(Td), but a twisted embedding associated to the cocyclegiven by the local action on Td. See Section 6 for details. We also note that theimage of G(F, F ′) in Gn,d does not intersect the amenable radical SVd,Sn−1

n , butdoes intersect the subgroup 1 o Aut(Td) (along a cocompact lattice of Aut(Td)).

The case n = 2 is particular as the group G2,d is actually a restricted wreathproduct, and in this situation the group G(F, F ′) is an irreducible cocompact latticein C2 ≀Vd

Aut(Td) = (⊕VdC2) o Aut(Td).

Applications. Recall that the property of being virtually simple is not invariantby quasi-isometry. Indeed the lattices constructed by Burger and Mozes in [BM00b]show that a virtually simple finitely generated group may have the same Cayleygraph as a product of two finitely generated free groups. Theorem 1.4 togetherwith simplicity results from [LB16] provide another illustration of this fact, namelyfinitely generated simple groups having the same Cayley graph as a wreath product.The wreath product construction is already known to be a source of examples offinitely generated groups whose algebraic properties are not reflected in their Cayleygraphs. Two wreath products B1 ≀ Γ and B2 ≀ Γ may have isometric or bi-LipschitzCayley-graphs, one being solvable or torsion free, while no finite index subgroup ofthe second has these properties [Dyu00]. The phenomenon exhibited in Theorem1.4 is nonetheless very different, in the sense that it provides finitely generatedgroups with isometric Cayley graphs such that one is a wreath product, but theother is simple (and hence not commensurable with a wreath product).

Recall that for finitely generated groups, being amenable is a quasi-isometryinvariant. By contrast, Theorem 1.4 implies:

Corollary 1.5. Among finitely generated groups, the property of having infiniteamenable radical is not invariant by quasi-isometry.

The examples from Theorem 1.4 simultaneously show that having an infiniteelliptic radical is also not invariant by quasi-isometry. Recall that the ellipticradical of a discrete group is the largest locally finite normal subgroup.

Recall that by a theorem of Eskin–Fisher–Whyte, any finitely generated group Γthat is quasi-isometric to a wreath product C ≀Z, where C is a finite group, must actproperly and cocompactly on a Diestel-Leader graph DL(m,m) [EFW07, EFW12,EFW13]. By the algebraic description of the isometry groups of these graphs givenin [BNW08] (see also [CFK12]), this implies in particular that Γ has a subgroupof index at most two that is (locally finite)-by-Z. By contrast, Theorem 1.4 showsthat this rigidity fails in the case of C ≀ Fk when k ≥ 2.

Questions. We end this introduction with two questions. Extreme proximalityis used in a crucial way at different stages of the proofs of Theorems 1.1 and 1.2.These results both fail without the extreme proximality assumption, simply becausethen the group itself may very well be a direct product. Putting aside these trivialcounter-examples, we do not know whether serious algebraic restrictions on a locally


compact group may be derived from the existence of a lattice with a continuousFurstenberg URS. In this direction, we find the following question natural:

Question 1.6. Does there exist Γ with a continuous Furstenberg URS which is alattice in a group G = G1 × G2 such both factors are non-discrete, and Γ has aninjective and dense projection to each factor ? What if we impose moreover thatΓ has trivial amenable radical ?

Theorem 1.4 presents a situation of a locally compact group G with two cocom-pact lattices Γ1,Γ2 ≤ G such that the stabilizer URS associated to the Γ1-action on∂spG is Rad(Γ1), while the stabilizer URS associated to the Γ2-action on ∂spG iscontinuous. Here ∂spG stands for the Furstenberg boundary of G; see §2.2. In theseexamples the group G splits as G = N oQ, where N is the amenable radical of G.The lattice Γ1 preserves this splitting, meaning that we have Γ1 = (N∩Γ1)o(Q∩Γ1)(and hence Γ1 does not act faithfully on ∂spG), while Γ2 has an injective projectionto Q. This naturally raises the following:

Question 1.7. Let G be a locally compact group with two lattices Γ1 and Γ2both acting faithfully on X = ∂spG. Is it possible that the Γ1-action on X istopologically free, but the Γ2-action on X is not topologically free ? Can thishappen with ∂spG = G/H being a homogeneous G-space ?

Note that by [Fur03, Prop. 7], the condition that Γ1 and Γ2 act faithfully on∂spG is equivalent to saying that Γ1 and Γ2 have trivial amenable radical. Recallthat topologically free means that there is a dense subset of points having trivialstabilizer (equivalently, the stabilizer URS is trivial).

Outline of proofs and organization. The article is organized as follows. In thenext section we introduce terminology and preliminary results about topologicalboundaries and extremely proximal actions. In Section 3 we establish the resultsabout uniformly recurrent subgroups that are used in later sections. In particularwe prove a certain gap property for URS’s coming from extremely proximal actions(Proposition 3.17). Combined with an observation about compact spaces withcomparable stabilizer URS’s (Proposition 3.9), we deduce that a locally compactgroup H with an amenable URS that comes from an extremely proximal action,and whose envelope is co-amenable in H, is boundary indivisible (Proposition 3.24).

The setting of Section 4 is that of a group admitting a non-topologically freeextremely proximal action. We establish intermediate results, notably concerningnormal subgroups (Proposition 4.6) and commensurated subgroups (Proposition4.12), and deduce non-embedding results for this class of groups (see Proposition4.10 and Corollary 4.13).

In Section 5 we use results from Section 3 together with Proposition 5.9 ofFurstenberg and prove Theorem 1.2. We then specify to discrete groups and givethe proof of Theorem 1.1. The proof essentially splits in two steps: the first one isthe application of Theorem 1.2 to obtain amenability of one factor, and the secondconsists in proving that under appropriate assumptions the amenable factor iscompact, using results from Section 4.

In Section 6 we consider groups acting on trees, and apply previous results ofthe article to this setting. After giving the proof of Corollary 1.3, we focus on thefamily of groups with prescribed local action G(F, F ′). We study boundaries of

10 ADRIEN LE BOUDEC

these groups, and use results from Section 3 in order to characterize the discretegroups within this family which are boundary indivisible (see Theorem 6.9). Thisincludes those which are virtually simple, but this also contain non-virtually simpleinstances. Finally we study lattice embeddings of these groups and give the proofof Theorem 1.4.

Acknowledgements. I am grateful to Alex Furman for pointing out Proposition5.9 to my attention, and to Uri Bader for enlightening discussion about the proof.I am also grateful to Pierre-Emmanuel Caprace, Yves Cornulier, Bruno Duchesne,Nicolás Matte Bon, Nicolas Monod and Pierre Pansu for interesting discussionsand comments related to this work. Finally I am indebted to Alain Valette for adecisive remark made in Neuchâtel in May 2015, which eventually led to Theorem1.4.

2. Preliminaries

2.1. Conventions and terminology. The letter G will usually refer to a topo-logical group, while Γ will denote a discrete group. The group of homeomorphicautomorphisms of G will be denoted Aut(G). Whenever G is a locally compactgroup, we will always assume that G is second countable.

The notation X will refer to a topological space. The letters X,Y will be re-served for compact spaces, and Z for a compact space equipped with an extremelyproximal group action. All compact spaces are assumed to be Hausdorff.

A space X is a G-space if G admits a continuous action G× X → X . The actionof G on X (or the G-space X ) is minimal if all orbits are dense. The G-space Xis said to be trivial if X is a one-point space.

If X is locally compact, we denote by Prob(X ) the set of all regular Borel proba-bility measures on X . The space of continuous compactly supported functions on Xis denoted CK(X ). Each µ ∈ Prob(X ) defines a linear functional on CK(X ), and weendow Prob(X ) with the weak*-topology: a net (µi) converges to µ if µi(f) → µ(f)for all f ∈ CK(X ). By Banach-Alaoglu theorem, Prob(X ) is relatively compact inCK(X )∗.

We denote by 2X the set of all closed subsets of X . The sets

O(K;U1, . . . , Un) =C ∈ 2X : C ∩K = ∅; C ∩ Ui = ∅ for all i

,

where K ⊂ X is compact and U1, . . . , Un ⊂ X are open, form a basis for theChabauty topology on 2X . Endowed with the Chabauty topology, the space 2X iscompact. We will freely identify X with its image in 2X by the natural inclusionx 7→ x. Note that when X is a G-space, so is 2X .

In the particular case where X = G is a locally compact group, the space Sub(G)of closed subgroups of G is closed in 2G. In particular Sub(G) is a compact space,on which G acts by conjugation. A uniformly recurrent subgroup (URS) ofG is a closed, G-invariant, minimal subset of Sub(G). The set of URS’s of G isdenoted URS(G). By extension we also say that a subgroup H ≤ G is uniformlyrecurrent if the closure of the conjugacy class of H in Sub(G) is minimal.

2.2. Topological boundaries. If X is a compact G-space, the action of G on Xis strongly proximal if the closure of any G-orbit in Prob(X) contains a Dirac


measure. Strong proximality is stable under taking products (with diagonal action)and continuous equivariant images (see e.g. [Gla76]).

We say that X is a boundary if X is both minimal and strongly proximal. Forevery topological group G, there exists a unique boundary ∂spG with the universalproperty that for any boundary X, there exists a continuous G-equivariant sur-jection ∂spG → X [Fur73, Prop. 4.6]. This universal space ∂spG is referred to asthe Furstenberg boundary of G. It is easy to verify that any amenable normalsubgroup N of G acts trivially on any G-boundary, so that ∂spG = ∂sp(G/N).

If G admits a cocompact amenable subgroup, then the Furstenberg boundaryis a homogeneous space ∂spG = G/H, and the G-spaces of the form G/L with Lcontaining H are precisely the G-boundaries [Fur73, Prop. 4.4]. The situation fordiscrete groups is quite different: as shown in [KK17] and [BKKO14], Furstenbergboundaries of discrete groups are always non-metrizable (unless trivial).

The following is a fundamental property of boundaries (see [Gla76, III.2.3]):

Theorem 2.1. Any convex compact G-space contains a boundary. In fact if Qis an irreducible convex compact G-space, then the action of G on Q is stronglyproximal, and the closure of extreme points of Q is a G-boundary.

Irreducible means that Q has no proper closed convex G-invariant subspace. Inparticular Theorem 2.1 has the following consequence ([Gla76, III.3.1]):

Theorem 2.2. A group G is amenable if and only if all G-boundaries are trivial,or equivalently ∂spG is trivial.

2.3. Extremely proximal actions. Let X be a compact G-space. A closed sub-set C of X is compressible if the closure of the G-orbit of C in the space 2X

contains a singleton x. Equivalently, for every neighbourhood U of x, there ex-ists g ∈ G such that g(C) ⊂ U . The action of G on X is extremely proximal ifevery closed subset C ( X is compressible. References where extremely proximalactions were considered include [Gla74, LS96, JR00, FST15, LBMB16].

We will make use of the following result, which is Theorem 2.3 from [Gla74]:

Theorem 2.3. Let X be a compact G-space, and assume X has at least threepoints. If the G-action on X is extremely proximal, then it is strongly proximal.

Examples of extremely proximal actions are provided by group actions on treesor hyperbolic spaces. If G ≤ Aut(T ) acts on T with no proper invariant subtreeand no finite orbit in T ∪∂T , then the action of G on ∂T is minimal and extremelyproximal; and if G acts coboundedly on a proper geodesic hyperbolic space X withno fixed point or fixed pair at infinity, then the G-action on the Gromov boundary∂X is minimal and extremely proximal.

These two situations are particular cases of the following more general result,that we believe is well-known. A homeomorphism g of a space X is hyperbolic ifthere exist ξ−, ξ+ ∈ X, called the endpoints of g, such that for all neighbourhoodsU−, U+ of ξ−, ξ+, for n large enough we have gn(X \U−) ⊂ U+ and g−n(X \U+) ⊂U−.

Proposition 2.4. If G acts on a compact space X with hyperbolic elements havingno common endpoints, and such that the set of endpoints of hyperbolic elements ofG is dense in X, then the action is minimal and extremely proximal.

12 ADRIEN LE BOUDEC

Proof. Let U ⊂ X be a non-empty open invariant subset. By our density assump-tion, there is g ∈ G hyperbolic whose attracting endpoint ξ+ belongs to U . So forevery x = ξ−, there is n > 0 such that gn(x) ∈ U since U is open, so we deduce thatU contains X \ ξ−. But the existence of hyperbolic elements with no commonendpoints ensures that G fixes no point of X, so finally U = X, i.e. the action isminimal.

Now if C ( X is a closed subset then again there is g ∈ G whose attractingendpoint is outside C, and C is compressible to the repealing endpoint of g.

Recent work of Duchesne and Monod shows that group actions on dendrites isalso a source of extremely proximal actions. Recall that a dendrite X is a compactmetrizable space such that any two points are the extremities of a unique arc.Duchesne and Monod show that if Γ acts on X with no invariant proper sub-dendrite, then there is a unique minimal closed invariant subset M ⊆ X and theΓ-action on M is extremely proximal. See the proof of Theorem 10.1 in [DM16].

Extremely proximal actions also play a prominent role in the context of group ac-tions on the circle. For any minimal action α : Γ → Homeo+(S1), either α(Γ) is con-jugated to a group of rotations, or α(Γ) has a finite centralizer CΓ in Homeo+(S1)and the action of Γ on the quotient circle CΓ\S1 is extremely proximal: see Ghys[Ghy01] and Margulis [Mar00]. We mention however that in all the examples ofcountable groups Γ with an action on S1 that is minimal and not topologically freethat we are aware of, the stabilizer URS is either non-amenable, or not known to beamenable. In particular we do not know any application of Theorem 1.1 to groupsacting on the circle.

In the sequel we will make use of the following easy lemma.

Lemma 2.5. Let G be a topological group, and H a subgroup of G such that thereis some compact subset K of G such that G = KH. Let X be a compact G-space,and C a closed subset of X that is compressible by G. Then C is compressible byH. In particular if the G-action on X is extremely proximal, then the H-action isextremely proximal.

Proof. By assumption there exists x ∈ X and (gi) such that gi(C) converges to xin 2X . If gi = kihi, by compactness of K we assume that (ki) converges to some k,and it follows that hi(C) converges to k−1x by continuity of G× 2X → 2X .

3. Uniformly recurrent subgroups

3.1. Generalities on uniformly recurrent subgroups. Let G be a locally com-pact group. For H,K ∈ URS(G), we write H 4 K when there exist H ∈ H andK ∈ K such that H ≤ K. This is equivalent to the fact that every H ∈ H iscontained in an element of K, and every K ∈ K contains an element of H, and therelation 4 is an order on URS(G). See e.g. §2.4 in [LBMB16].

For simplicity the URS N associated to a closed normal subgroup N of Gwill still be denoted N . In particular N 4 H (resp. H 4 N) means that N iscontained in (resp. contains) all the elements of H. By the trivial URS we meanthe URS corresponding to the trivial subgroup 1. We warn the reader that inthis terminology the URS corresponding to a non-trivial normal subgroup N istrivial as a G-space (it is a one-point space), but is not trivial as a URS.


Let X,Y be compact G-spaces. We say that X is a factor of Y , and Y is anextension of X, if there exists a continuous equivariant map Y → X that is onto.If π : Y → X is a continuous equivariant map, we say that π is almost 1-1 if theset of y ∈ Y such that π−1(π(y)) = y is dense in Y . When moreover π is ontowe say that Y is an almost 1-1 extension of X.

We now recall the definition of the stabilizer URS associated to a minimal actionon a compact space. If X is a compact G-space and x ∈ X, we denote by Gx thestabilizer of x in G.

Definition 3.1. If X is a compact G-space, we denote by X0 ⊂ X the set of pointsat which Stab : X → Sub(G), x 7→ Gx, is continuous.

Upper semi-continuity of the map Stab and second countability of G imply thatX0 is a dense subset of X (indeed if (Un) is a basis of the topology on Sub(G) andXn is the set of x ∈ X such that Gx ∩ Un = ∅, which is closed, one verifies thatStab is continuous on ∩n(∂Xn)c). Following [GW15], we denote

X = cls (x,Gx) : x ∈ X0 ⊂ X × Sub(G),and

SG(X) = cls Gx : x ∈ X0 ⊂ Sub(G),where cls stands for the closure in the ambient space. We have the obvious inclu-sions

X ⊆ X∗ := cls (x,Gx) : x ∈ Xand

SG(X) ⊆ S∗G(X) := cls Gx : x ∈ X .

We denote by ηX and πX the projections from X × Sub(G) to X and Sub(G)respectively.

Proposition 3.2 (Prop. 1.2 in [GW15]). If X is a minimal compact G-space, thenηX : X → X is an almost 1-1 extension, and X and SG(X) are the unique minimalclosed G-invariant subsets of respectively X∗ and S∗

G(X).

Definition 3.3. SG(X) is the stabilizer URS associated to the G-action on X.The action of G on X is topologically free if SG(X) is trivial, i.e. SG(X) = 1.

Remark 3.4. When G is not assumed second countable, in general X0 is no longerdense in X. However it is still possible to define the stabilizer URS associated toa minimal action on a compact space; see the discussion in [MBT17, p1].

In the sequel we will sometimes use the following version of Proposition 3.2.

Proposition 3.5. Let X be compact G-space, and let H ≤ G be a subgroup actingminimally on X. Then H acts minimally on X and SG(X), and X and SG(X)are the unique minimal closed H-invariant subsets of X∗ and S∗

G(X).

Proof. Let Y ⊆ X∗ closed and H-invariant. Since X is a factor of X∗ and H actsminimally on X, for every x ∈ X0 there exists L ∈ Sub(G) such that (x, L) ∈ Y .But for x ∈ X0, the fact that (x, L) belongs to X∗ forces L to be equal to Gx

by definition of X0, and it follows that X ⊆ Y . Moreover H acts minimally onX since ηX : X → X is an almost 1-1 extension and minimality is preserved bytaking almost 1-1 extensions (if π : X1 → X2 is almost 1-1 and if C ⊆ X1 is a

14 ADRIEN LE BOUDEC

closed subset such that π(C) = X2, then C = X1). So the statements for X andX∗ are established, and the same hold for SG(X) and S∗

G(X) since these are factorsof X and X∗.

3.2. Envelopes. Let G be a locally compact group and H ∈ URS(G).

Definition 3.6. The envelope Env(H) of H is the closed subgroup of G generatedby all the subgroups H ∈ H.

By definition Env(H) is the smallest closed subgroup of G such that H ⊂Sub(Env(H)). Note that Env(H) is a normal subgroup of G, and is actually thesmallest normal subgroup such that H 4 Env(H).

Let Γ be a discrete group, X a compact Γ-space and X0 the domain of continuityof the map Stab. It is a classical fact that X0 consists of those x ∈ X such that forevery γ ∈ Γx, there exists U neighbourhood of x that is fixed by γ (see e.g. [Vor12,Lem. 5.4] for a proof). For x ∈ X, we will denote by Γ0

x ≤ Γx the set of elementsfixing a neighbourhood of x, so that x ∈ X0 if and only if Γx = Γ0

x.

Lemma 3.7. Let Γ be a countable discrete group, X a compact minimal Γ-space,n ≥ 1 and γ1, . . . , γn ∈ Γ. The following are equivalent:

(i) ∩Fix(γi) has non-empty interior;(ii) there is x ∈ X such that γi ∈ Γ0

x for all i;(iii) same as in (ii) but with x ∈ X0;(iv) there is H ∈ SΓ(X) such that γi ∈ Hfor all i.

In particular Env(SΓ(X)) is generated by the elements γ ∈ Γ such that Fix(γ) hasnon-empty interior.

Proof. It is clear that (i) and (ii) are equivalent. Also (iii) clearly implies (ii),and (ii) also implies (iii) by density of X0 in X. Finally (iii) implies (iv) sinceΓ0

x ∈ SΓ(X) for x ∈ X0, and (iv) implies (iii) by density of the set of Γ0x, x ∈ X0,

in SΓ(X).

3.3. G-spaces with comparable stabilizer URS’s. We recall the notion ofdisjointness from [Fur67a]. Two compact G-spaces X,Y are disjoint if wheneverX,Y are factors of a compact G-space Ω via φX : Ω → X and φY : Ω → Y , thenthe map (φX , φY ) : Ω → X × Y is surjective. When X,Y are minimal G-spaces,this is equivalent to saying that the product X×Y remains minimal [Fur67a, Lem.II.1].

The following lemma presents a situation which easily implies disjointness:

Lemma 3.8. Let X,Y be minimal compact G-spaces such that there exists y0 ∈ Ysuch that Gy0 acts minimally on X. Then X and Y are disjoint.

Proof. This is clear: if W is a closed invariant subset of X×Y , then by minimalityof Y there exists x0 ∈ X such that (x0, y0) ∈ W . Since Gy0 acts minimally on Xwe deduce that W contains X × y0, and by minimality of Y it follows that W isequal to X × Y .

The following proposition will be used notably in Proposition 3.24.


Proposition 3.9. Let X,Y be compact minimal G-spaces, and write H = SG(X)and K = SG(Y ). Suppose that H 4 K. Then X and Y can be disjoint only ifEnv(H) 4 K.

In particular if SG(X) = SG(Y ) and this URS is not a point, then X and Y arenot disjoint.

Proof. Using notation from Proposition 3.2, we have almost 1-1 extensions ηX :X → X and ηY : Y → Y , and we write η = ηX × ηY : X × Y → X × Y , andπ = πX × πY : X × Y → H × K. The set W ⊆ H × K of pairs (H,K) such thatH ≤ K is non-empty by assumption, is clearly G-invariant, and is easily seen to beclosed. If W is a proper subset of H × K then π−1(W ) is a proper subset of X × Ysince π is a factor, and it follows that η

(π−1(W )

)is a closed G-invariant subset of

X × Y that is proper since η is almost 1-1. This contradicts disjointness of X andY . Therefore we have W = H × K. This means that for a fixed K ∈ K we haveH ≤ K for every H ∈ H, and hence Env(H) 4 K.

3.4. Action of a URS on a G-space. In this paragraph G still denote a locallycompact group, and X is a compact G-space. Given H ∈ URS(G), we study theproperties of the action of elements on H on the space X.

The proof of the following lemma is an easy verification, and we leave it to thereader.

Lemma 3.10. If X is a compact G-space, H ∈ Sub(G) |H fixes a point in X isa closed G-invariant subset of Sub(G).

In particular the following definition makes sense.

Definition 3.11. Let X be a compact G-space, and H ∈ URS(G). We say thatH fixes a point in X if for some (all) H ∈ H, there is x ∈ X such that h(x) = xfor all h ∈ H.

Lemma 3.12. Let X be a compact G-space, Y ⊆ X a closed invariant subset ofX, and H ∈ URS(G). If there exists H ∈ H fixing x ∈ X such that Gx ∩ Y = ∅,then H fixes a point in Y .

Proof. By assumption there exist (gi) and y ∈ Y such that gi(x) converges to y. IfK ∈ H is a limit point of (Hgi) (which exists by compactness), then K fixes y byupper semi-continuity of the stabilizer map.

Lemma 3.12 implies the following:

Lemma 3.13. If X is a compact G-space containing a unique minimal closed G-invariant subset Xmin ⊂ X (e.g. X is proximal), and if H ∈ URS(G) fixes a pointin X, then H fixes a point in Xmin.

Proposition 3.14. Let Z be a compact G-space that is extremely proximal, andH ∈ URS(G). Then either H fixes a point in Z, or all H ∈ H act minimally onZ.

Proof. If there exist H ∈ H and a non-empty closed subset C ( Z that is invariantby H, then we may apply Lemma 3.12 to the space X = 2Z , the subspace Y = Zand the point x = C, and we deduce that H fixes a point in Z.

16 ADRIEN LE BOUDEC

Recall that given a compact G-space X, S∗G(X) stands for the closure in Sub(G)

of the set of subgroups Gx, x ∈ X.

Lemma 3.15. Let X be a compact G-space. Assume that K ≤ G is a closedsubgroup of G which acts minimally on X and such that there exists H ∈ S∗

G(X)with H ≤ K. Then Env(SG(X)) ≤ K.

Proof. Since K acts minimally on X, the closure of the K-orbit of H in S∗G(X)

contains SG(X) according to Proposition 3.5. Since H ∈ Sub(K) and Sub(K) isa closed subset of Sub(G), we deduce that SG(X) ⊂ Sub(K), and in particularEnv(SG(X)) ≤ K.

Definition 3.16. Let H ∈ URS(G). We say that H comes from an extremelyproximal action if there exists a compact G-space Z that is minimal and ex-tremely proximal, and such that SG(Z) = H.

It was shown in [LBMB16] that for a discrete group Γ with a non-trivial URSH coming from an extremely proximal action, any non-trivial K ∈ URS(Γ) mustbe “relatively large” with respect to H (see [LBMB16, Th. 3.10] for a precisestatement). Appropriate assumptions on Γ and H further imply that H 4 K forevery non-trivial K ∈ URS(Γ) [LBMB16, Cor. 3.12]. The following propositiongoes in the opposite direction by considering URS’s larger than H.

Proposition 3.17. Let H ∈ URS(G) that comes from an extremely proximal ac-tion. Let K ∈ URS(G) such that H 4 K and H = K. Then Env(H) 4 K.

Proof. Let Z be a compact G-space that is minimal and extremely proximal andsuch that SG(Z) = H. Fix K ∈ K, and assume that K does not act minimallyon Z. According to Proposition 3.14 this implies that the URS K fixes a pointin Z, i.e. K 4 H. Since moreover H,K satisfy H 4 K by assumption, we deducethat H = K, which is a contradiction. Therefore K acts minimally on Z. Sincemoreover there exists H ∈ H such that H ≤ K, we are in position to apply Lemma3.15, from which the conclusion follows.

It should be noted that Proposition 3.17 is false without the extreme proximalityassumption, as in general there are plenty of URS’s between H and Env(H).

Lemma 3.18. Let H ∈ URS(G) that comes from an extremely proximal action.Then Env(H) acts minimally on H.

Proof. Let Z be a compact G-space that is minimal and extremely proximal andsuch that SG(Z) = H, and let N = Env(H). Without loss of generality we mayassume that H is not a point, since otherwise there is nothing to prove. This ensuresthat N acts non-trivially on Z. By extreme proximality N must act minimally onZ (see Lemma 4.2), and therefore also on H by Proposition 3.5.

Remark 3.19. The extreme proximality assumption cannot be removed in Lemma3.18. Indeed it is not true in general that, given H ∈ URS(G), H remains a URSof Env(H). Indeed, as explained in [GW15], any minimal subshift on two lettersgives rise to a URS H of the lamplighter group G = C2 ≀Z, such that H is containedin the Chabauty space Sub(L) of the base group L = ⊕C2. In particular Env(H)lies inside the abelian group L, and it follows that Env(H) acts trivially on H.


Proposition 3.20. Let H ∈ URS(G) that comes from an extremely proximal ac-tion, and assume H is not a point. Then:

(a) The action of G on H gives rise to the same URS, i.e. SG(H) = H.(b) If moreover H comes from a faithful extremely proximal action, then the

action of G on H is faithful.

Proof. Write K = SG(H). By definition we have H 4 K. Argue by contradictionand suppose H = K. Then applying Proposition 3.17, we deduce that Env(H) actstrivially on H. But Env(H) also acts minimally on H by Lemma 3.18, so we deducethat H must be a point, a contradiction. This shows (a).

For (b), arguing as in the proof of Lemma 3.18 we see that any non-trivial normalsubgroup N of G acts minimally on H. Since H is not a point, we have in particularthat N acts non-trivially on H. Remark 3.21. Proposition 3.20 implies that, as far as our interest lies inside theURS associated to a minimal and extremely proximal action (and not the spaceZ itself), there is no loss of generality in assuming that (G,Z) is a sub-system of(G, Sub(G)). See also Remark 3.27.

3.5. Amenable URS’s. Recall that we say that H ∈ URS(G) is amenable if everyH ∈ H is amenable. The following lemma already appeared in [LBMB16, Prop.2.21].

Lemma 3.22. If H ∈ URS(G) is amenable and X is a G-boundary, then H 4SG(X).

Proof. Since H is amenable, H must fix a point in the compact G-space Prob(X).Now X is the unique minimal G-invariant subspace of Prob(X) since X is a G-boundary, so by Lemma 3.13 we have that H fixes a point in X, i.e. H 4 SG(X). Proposition 3.23. Let X be a compact minimal G-space such that H = SG(X)is amenable, and let Y be a G-boundary such that X and Y are disjoint. ThenEnv(H) acts trivially on Y .

In particular if Env(H) is co-amenable in G, a non-trivial G-boundary is neverdisjoint with X.

Proof. The fact that Env(H) must act trivially on Y follows by applying Lemma3.22 and Proposition 3.9. Since an amenable group has no non-trivial boundary,the second statement follows.

Proposition 3.23 says that when G admits an amenable URS whose envelope isco-amenable, a non-trivial G-boundary is never disjoint with X. This conclusion isnot satisfactory for our concerns as it depends on the choice of a space X and notonly on G. Although there is no hope to get a better conclusion in full generality,the next result, which will play an important role in Section 5, will remove thisdependence under an extreme proximality assumption.

We recall from the introduction that we say that G is boundary indivisible iftwo non-trivial G-boundaries are never disjoint.

Proposition 3.24. Assume that G admits an amenable H ∈ URS(G) that comesfrom an extremely proximal action, and let X be a non-trivial G-boundary.

(a) Either SG(X) = H, or Env(H) acts trivially on X.

18 ADRIEN LE BOUDEC

(b) Assume that Env(H) is co-amenable in G. Then SG(X) = H, and G isboundary indivisible.

Proof. (a). Since H is amenable, we have H 4 SG(X) by Lemma 3.22. Now if weassume H = SG(X), then according to Proposition 3.17 we have Env(H) 4 SG(X),which exactly means that Env(H) acts trivially on X.

(b). If SG(X) = H then the action of G on X factors through an action ofG/Env(H) by (a). But by assumption the latter is amenable, so has no non-trivialboundaries. So it follows that X is trivial, a contradiction. Therefore all non-trivialG-boundaries have the same stabilizer URS H. Since moreover H cannot be a point(because otherwise G would be amenable), the fact that G is boundary indivisiblefollows from Proposition 3.9.

For a countable group Γ, the Furstenberg URS of Γ is the stabilizer URSassociated to the action of Γ on its Furstenberg boundary. We refer to [LBMB16]for the proof of the following properties.

Proposition 3.25. Let Γ be a countable group, and AΓ its Furstenberg URS. Thenthe following hold:

(a) AΓ is amenable, and H 4 AΓ for every amenable H ∈ URS(Γ).(b) If X is a Γ-boundary, then AΓ 4 SΓ(X). If moreover there is x ∈ X such

that Γx is amenable, then AΓ = SΓ(X).(c) AΓ is invariant under Aut(Γ).

Proposition 3.26. Let Γ be a countable group, and let Λ = Env(AΓ) be the enve-lope of the Furstenberg URS of Γ. Then Λ acts minimally on AΓ, and AΓ = AΛ.

Proof. The conjugation action of Γ on the normal subgroup Λ = Env(AΓ) inducesa map Γ → Aut(Λ). Since AΛ is invariant under Aut(Λ) by Proposition 3.25, it isin particular Γ-invariant. Moreover the action of Γ on AΛ is clearly minimal sinceit is already the case for Λ. Therefore AΛ is an amenable URS of Γ, so it followsthat AΛ 4 AΓ since AΓ is larger than any amenable URS of Γ. On the other handAΓ is a closed and Λ-invariant subset of Sub(Λ) consisting of amenable subgroups,so by the domination property applied to AΛ we must have AΓ 4 AΛ. Equalityfollows. Remark 3.27. When Env(AΓ) is co-amenable in Γ, the fact that AΓ comes froma faithful and extremely proximal action is equivalent to saying that the Γ-actionon AΓ is faithful and extremely proximal. The direct implication is consequenceof Proposition 3.20, and the converse follows from Proposition 3.24. This gives usan intrinsic reformulation of the assumption of Theorem 1.1 inside the Chabautyspace of Γ.

4. Extremely proximal actions

If X is a Hausdorff Γ-space and U ⊂ X , we denote by ΓU the set of elements ofΓ acting trivially on X \U . We say that the action of Γ on X is micro-supportedif ΓU is non-trivial for every non-empty open set U .

We will need the following easy lemma.

Lemma 4.1. Assume that the action of Γ on X is micro-supported, and let U bea non-empty open set. Then ΓU is not solvable.


Proof. Assume that Λ is a subgroup of ΓU whose action on U is micro-supported,and let V be a non-empty open subset of U . By assumption there exists a non-trivial λ1 ∈ ΛV , so that we may find an open set W ⊂ V such that W and λ1(W )are disjoint. For λ2 ∈ ΛW , the commutator [λ1, λ2] coincides with λ−1

2 on W , andis therefore non-trivial provided that λ2 is non-trivial. It follows by induction thatif ΓU,n is the n-th term of the derived series of ΓU , then the action of ΓU,n on U ismicro-supported. In particular ΓU,n is never trivial, and ΓU is not solvable.

In this section we will consider the following setting:(EP) Γ is a discrete group, Z is a compact Γ-space, and the action of Γ on Z is

faithful, minimal and extremely proximal. In order to avoid trivialities, we assumethat Z has at least three points.

Unless specified otherwise, in the remaining of this section Γ and Z will beassumed to satisfy (EP). Our goal is to derive various properties on the group Γthat will be used in later sections.

Lemma 4.2. Let N ≤ Homeo(Z) be a non-trivial subgroup that is normalized byΓ. Then N acts minimally and does not fix any probability measure on Z.

Proof. Assume there exists C ( Z that is closed and N -invariant. Since C iscompressible and N is normalized by Γ, wee see that N has a fixed point in Z. Nowthe set of N -fixed points is Γ-invariant, so it has to be the entire Z by minimality,and N is trivial. The same argument shows the absence of N -invariant probabilitymeasure on Z, since an extremely proximal action is also strongly proximal byTheorem 2.3.

In all this section the terminology topologically free (see Definition 3.3) has tobe understood with Γ viewed as a discrete group. Therefore that the action is nottopologically free means that there exists γ = 1 which acts trivially on a non-emptyopen subset of Z.

Lemma 4.3. If the action of Γ on Z is not topologically free, then it is micro-supported.

Proof. Let U be a non-empty open subset of Z. Let γ be a non-trivial elementsuch that there is a non-empty open set V on which γ acts trivially, and let g ∈ Γsuch that g(X \ V ) ⊂ U . Then the non-trivial element gγg−1 acts trivially outsideU , so ΓU is non-trivial. Definition 4.4. Let Γ0 be the subgroup of Γ generated by the elements γ ∈ Γsuch that Fix(γ) has non-empty interior.

Remark 4.5. When Γ is a countable group, Γ0 is also equal to the envelope of theURS SΓ(Z) by Lemma 3.7.

Recall that the monolith Mon(Γ) is the intersection of all non-trivial normalsubgroups of Γ. We say Γ is monolithic if Mon(Γ) is non-trivial.

Proposition 4.6. Assume that the action of Γ on Z is not topologically free. Thenthe following hold:

(a) The commutators [γ1, γ2], where Fix(γ1) ∩ Fix(γ2) has non-empty interior,generate [Γ0,Γ0].

20 ADRIEN LE BOUDEC

(b) Γ is monolithic, and one has Mon(Γ) = [Γ0,Γ0].(c) Any non-trivial normal subgroup of Γ has trivial centralizer.(d) If the action of [Γ0,Γ0] on Z is extremely proximal, then [Γ0,Γ0] is a simple

group.(e) Γ is virtually simple if and only if Γ0 has finite index in Γ and finite abelian-

ization.Proof. (a) Denote by N the subgroup generated by the set of [γ1, γ2], where γ1, γ2act trivially on a common open set. We show that for every g, h fixing non-emptyopen sets U, V , the commutator [g, h] belongs to N . Since Γ0 is generated by allthese elements g, h, this will show that Γ0/N is abelian. Hence [Γ0,Γ0] ≤ N , andthe other inclusion is clear.

First note that N is not trivial by Lemmas 4.1 and 4.3. Therefore N actsminimally on Z according to Lemma 4.2, and we may find s ∈ N such that theopen set W = U ∩ s(V ) is non-empty. Since g and hs fix W by construction, wehave [g, hs] ∈ N . But since s ∈ N , we deduce that [g, h] = [h, s]g[g, hs][s, h] is aproduct of three elements of N , and hence belongs to N , as desired.

(b) We shall show that any non-trivial normal subgroup N contains [Γ0,Γ0].Since [Γ0,Γ0] is itself a non-trivial normal subgroup, this will prove that it is themonolith of Γ. By a classical commutator manipulation (see e.g. Lemma 4.1 from[Nek13]), there exists an open set U such that N contains the derived subgroupof ΓU . Now let γ1, γ2 fixing an open set V . If γ is such that γ(Z \ V ) ⊂ U , thenγγ

1 , γγ2 are supported inside U , so that [γγ

1 , γγ2 ] = [γ1, γ2]γ is contained in N . Since

N is normal, [γ1, γ2] ∈ N . Now all these elements generate [Γ0,Γ0] by (a), hencethe conclusion.

(c) If N is a normal subgroup of Γ, then so is CΓ(N). Therefore they cannot beboth non-trivial, because otherwise the intersection would be abelian and wouldcontain [Γ0,Γ0] by the previous paragraph, a contradiction.

(d) If the action of [Γ0,Γ0] on Z is extremely proximal, then according to (b)the monolith N of [Γ0,Γ0] is non-trivial. Since N is characteristic in [Γ0,Γ0], Nis normal in Γ, and hence contains [Γ0,Γ0] by (b). So N = [Γ0,Γ0] and [Γ0,Γ0] issimple.

(e) For Γ to be virtually simple it is clearly necessary that the normal subgroup[Γ0,Γ0] has finite index in Γ. Conversely, if this condition holds then the action of[Γ0,Γ0] on Z is extremely proximal (Lemma 2.5), and [Γ0,Γ0] is simple by (d). Definition 4.7. Let X be a topological space, and let Γ be a group acting on X .A non-empty open set Ω ⊂ X is wandering for Γ if the translates γ(Ω), γ ∈ Γ,are pairwise disjoint. We say that Ω is wandering for γ if it is wandering for ⟨γ⟩.Proposition 4.8. Let Γ and Z as in (EP). Then there exist an open set Ω and anon-abelian free subgroup F2 = ∆ ≤ Γ such that Ω is wandering for ∆.Proof. Following Glasner [Gla74], we consider pairwise disjoint non-empty opensets U−, U+, V−, V+, and elements a, b ∈ Γ such that a(Z\U−) ⊂ U+ and b(Z\V−) ⊂V+. Let W = U− ∪ U+ ∪ V− ∪ V+. It follows from a ping-pong argument than anynon-trivial reduced word in the letters a, b sends the complement of W inside W ,so that the subgroup ∆ generated by a, b is free [Gla74, Th. 3.4].

Upon reducing W if necessary, we may find an open set Ω such that Ω ∩W = ∅and a(Ω) ⊂ U+, a−1(Ω) ⊂ U−, b(Ω) ⊂ V+ and b−1(Ω) ⊂ V−. Induction on the


word length shows that if the left-most letter of γ ∈ ∆ is respectively a, a−1, b, b−1,then γ(Ω) lies respectively inside U+, U−, V+, V−. In particular Ω ∩ γ(Ω) is emptysince Ω is disjoint from W , so Ω is wandering for ∆. Proposition 4.9. Retain notations (EP). Then the wreath product ΓU ≀F2 embedsinto Γ for every open subset U ( Z that is not dense.

Proof. Let Ω and ∆ as in Proposition 4.8, and let Λ be the subgroup of Γ generatedby ΓΩ and ∆. Since Ω is wandering for ∆, all the conjugates γΓΩγ

−1 pairwisecommute, and it follows that Λ is isomorphic to ΓΩ ≀ F2. Now if U is an in thestatement, by extreme proximality the group ΓU is isomorphic to a subgroup ofΓΩ, hence the conclusion.

The argument in the following proof is borrowed from [Cap16].

Proposition 4.10. In the setting (EP), if the action of Γ on Z is not topologicallyfree, then Γ cannot have any faithful linear representation.

Proof. Let U be an open subset of Z. By Lemmas 4.1 and 4.3, we may find anon-abelian finitely generated subgroup B inside ΓU . Now if we choose U smallenough, it follows from Proposition 4.9 that the finitely generated group Λ = B ≀F2is isomorphic to a subgroup of Γ. Since Λ is not residually finite [Gru57], it admitsno faithful linear representation by Malcev’s theorem [Mal40], and a fortiori thesame is true for Γ.

Recall that a subgroup Λ of a group Γ is commensurated if all conjugates ofΛ are commensurable, where two subgroups are commensurable if the intersectionhas finite index in both.

The beginning of the argument in the proof of the following proposition alreadyappeared in [LBW16]. The idea is to extend classical techniques for normal sub-groups to certain commensurated subgroups.

Proposition 4.11. Retain notation from (EP), and assume that the action of Γon Z is not topologically free. If Λ is a commensurated subgroup of Γ such thatthere exists an element of Λ admitting a wandering open set, then Λ contains themonolith Mon(Γ).

Proof. Let λ ∈ Λ admitting a wandering open set Ω. We shall first prove that[ΓΩ,ΓΩ] is contained in Λ. Let g, h ∈ ΓΩ, and let also n ≥ 1. Since Ω is wanderingwe have Ω ∩ λn(Ω) = ∅. It follows that the commutator [g, λn] is trivial outsideΩ ∪ λn(Ω), and coincides with g on Ω and with λng−1λ−n on λn(Ω). Thereforeits commutator with h is trivial outside Ω, and is coincides with [g, h] on Ω. Butsince g, h ∈ ΓΩ, the elements [[g, λn], h] and [g, h] actually coincide everywhere, i.e.[[g, λn], h] = [g, h].

Now since Λ is commensurated in Γ, there exists n0 ≥ 1 such that [[g, λn0 ], h]belongs to Λ. Applying the previous argument with n = n0, we deduce that [g, h]belongs to Λ.

In order to prove the statement, it is enough to prove that [ΓC ,ΓC ] is containedin Λ for every closed subset C ( Z according to Proposition 4.6. So let C a properclosed subset of Z. By minimality and extreme proximality, there is γ ∈ Γ suchthat γ(C) ⊂ Ω. Fix such a γ, and choose some integer n1 ≥ 1 such that γ−1λn1γbelongs to Λ. Set λ′ = γ−1λn1γ and Ω′ = γ−1(Ω). This Ω′ is wandering for λ′ and

22 ADRIEN LE BOUDEC

λ′ ∈ Λ, so [ΓΩ′ ,ΓΩ′ ] is contained in Λ by the first paragraph. Since C ⊂ Ω′, wehave ΓC ≤ ΓΩ′ , and the proof is complete.

Proposition 4.12. Assume that Z is a compact Γ-space and the action of Γ onZ is faithful, minimal, extremely proximal and not topologically free. Then thereexists a free subgroup F2 ≤ Γ such that for every commensurated subgroup Λ ≤ Γnot containing the monolith Mon(Γ), we have F2 ∩ Λ = 1.

Proof. Let F2 be a free subgroup of Γ as in the conclusion of Proposition 4.8. IfΛ ≤ Γ is a commensurated subgroup such that F2 ∩ Λ = 1, then in particular Λcontains an element admitting a wandering open set. So by Proposition 4.11 wehave Mon(Γ) ≤ Λ. This shows that every commensurated subgroup not containingMon(Γ) intersects F2 trivially.

Corollary 4.13. Assume that Z is a compact Γ-space and the action of Γ on Zis faithful, minimal, extremely proximal and not topologically free. If G is a locallycompact amenable group whose connected component G0 is a Lie group, then thereexists no injective homomorphism Γ → G.

Proof. Argue by contradiction and assume Γ embeds in G. Let U be an opensubgroup of G containing G0 as a cocompact subgroup. Such a U is commensuratedin G, so the subgroup Γ ∩ U is commensurated in Γ. If there exists U such thatΓ ∩U does not contain Mon(Γ), then according to Proposition 4.12 we may find anon-abelian free subgroup F2 ≤ Γ such that F2 ∩ U = 1. In particular G containsthe non-amenable group F2 as a discrete subgroup, which contradicts amenabilityof G. Therefore Mon(Γ) is contained in U for every choice of U . Since compactopen subgroups form a basis at 1 in G/G0 by van Dantzig’s theorem, it follows thatMon(Γ) actually lies inside G0. Now since G0 is a connected Lie group, the groupAut(G0) is linear, so the map G → Aut(G0) induced by the conjugation action ofG on G0 is not injective in restriction to Γ by Proposition 4.10. Therefore this mapmust vanish on Mon(Γ), which means that Mon(Γ) actually lies inside the centerof G0. In particular Mon(Γ) is abelian, which contradicts Proposition 4.6.

5. The proofs of Theorems 1.1 and 1.2

5.1. Boundary-minimal subgroups. In this paragraph we consider the follow-ing property:

Definition 5.1. Let G be a topological group, and L a closed subgroup of G.We say that L is boundary-minimal if there exists a non-trivial G-boundary onwhich L acts minimally.

It should be noted that being boundary-minimal does not prevent L from beingamenable. For instance the action of Thompson’s group T on the circle S1 isa boundary action, and the abelian subgroup of T consisting of rotations actsminimally on S1. Other examples may be found among the groups acting on treesconsidered in §6.2, where the stabilizer of a vertex is an amenable subgroup actingminimally on the ends of the tree.

In the sequel we will mainly focus on the case when L is normal in G, or moregenerally when L belongs to a URS (see Proposition 5.8). By contrast with theprevious examples, a normal boundary-minimal subgroup is never amenable, as


a normal amenable subgroup of G acts trivially on any G-boundary. Recall thatFurman showed [Fur03, Prop. 7] (see also Caprace–Monod [CM14, Prop. 3]) thatif N is non-amenable normal subgroup of a locally compact group G, there alwaysexists a G-boundary on which N acts non-trivially. This naturally raises the ques-tion whether any non-amenable normal subgroup of G is boundary-minimal. Wedo not know the answer to this question. While the case of discrete groups is easilysettled (see below), the situation for non-discrete groups seems to be more delicate.

We recall the following result of Furstenberg (see [Gla76, II.4.3]):

Theorem 5.2. Let G be a topological group, and denote by φ : G → Homeo(∂spG)the action of G on ∂spG. Then there exists a homomorphism ψ : Aut(G) →Homeo(∂spG) such that ψ Inn = φ, where Inn(G) ≤ Aut(G) is the group of innerautomorphisms of G.

In particular when N is a normal subgroup of a group G, the map G → Aut(N)coming from the conjugation action of G on N induces an action of G on ∂spN ,which factors through G/CG(N).

Note that this result readily answers the above question for discrete groups,by showing that the boundary-minimal normal subgroups of G are exactly thenon-amenable normal subgroups of G. Indeed if N is non-amenable then ∂spNis a non-trivial space, N acts minimally on ∂spN , and ∂spN is a G-boundary byTheorem 5.2. However the argument does not carry over for arbitrary groups, asin general the G-action on ∂spN is not continuous.

Proposition 5.3. Let G be a locally compact group, and N a closed normal non-amenable subgroup. Assume that at least one of the following holds true:

(a) N · CG(N) is open in G (e.g. when N is a direct factor of G);(b) N is cocompact in G;(c) there exists H ∈ URS(N) that is not a point, invariant by Aut(N), and that

is an N -boundary (e.g. if N has a closed cocompact amenable subgroup).Then N is boundary-minimal in G.

Proof. Condition (a) ensures that the image of N in G/CG(N) is open. Thereforethe G-action on ∂spN given by Theorem 5.2 is continuous because the N -actionis, and we deduce that ∂spN is a non-tivial G-boundary. If (b) holds, then N actsminimally on any G-boundary [Gla76, II.3.2]. Finally the verification of case (c)is straightforward since G → Aut(N) is continuous [HR79, Th. 26.7] and Aut(N)acts continuously on Sub(N).

5.2. Weakly co-amenable subgroups. In this paragraph we consider the fol-lowing weakening of the notion of co-amenability.

Definition 5.4. Let G be a topological group, and H a subgroup of G. We say thatH is weakly co-amenable in G if whenever Q is a non-trivial convex compactG-space in which H fixes a point, Q is not irreducible.

The following properties readily follow from the definition.

Proposition 5.5. Let K ≤ H ≤ G be subgroups of G.(i) If H ≤ G is co-amenable then H is weakly co-amenable.

24 ADRIEN LE BOUDEC

(ii) For a normal subgroup N G, weakly co-amenable is equivalent to co-amenable.

(iii) If H ≤ G is amenable and weakly co-amenable in G, then G is amenable.(iv) If φ : G → G′ is continuous with dense image and H ≤ G is weakly co-

amenable, then φ(H) is weakly co-amenable in G′.(v) If K is weakly co-amenable in G, then H is weakly co-amenable in G.

(vi) If K is co-amenable in H and H is weakly co-amenable in G, then K isweakly co-amenable in G.

Proof. (i) If Q is non-trivial convex compact G-space with H-fixed points, thenthere is a G-fixed point by co-amenability of H in G, so Q is not irreducible.

(ii) If N G is not co-amenable, there is a convex Q such that Fix(N) is non-empty but Fix(G) is empty. Since N is normal Fix(N) is G-invariant, so that byZorn’s lemma Fix(N) contains an irreducible convex G-space, which is non-trivialsince Fix(G) is empty. This shows N is not weakly co-amenable.

The proofs of (iii), (iv), (v) and (vi) are similar verifications, and we leave themto the reader. Remark 5.6. As for co-amenability, it is natural to wonder whether weak co-amenability of K in G implies weak co-amenability of K in H. In view of (ii),the same counter-examples given in [MP03] show that the answer is negative ingeneral.

By the correspondence between irreducible convex compact G-spaces and G-boundaries, weak co-amenability admits the following characterization:

Proposition 5.7. A subgroup H ≤ G is weakly co-amenable in G if and only iffor every non-trivial G-boundary X, there is no probability measure on X that isfixed by H.

Proof. Follows from Theorem 2.1. The following shows how weak co-amenability naturally appears for boundary

indivisible groups (see also Proposition 6.17).

Proposition 5.8. Let G be a boundary indivisible locally compact group, and L aclosed subgroup of G that is boundary-minimal and uniformly recurrent. Then L isweakly co-amenable in G.

Proof. Write H for the closure of LG in Sub(G), which is a URS by assumption.Let X be a non-trivial G-boundary on which L acts minimally, and let Y be aG-boundary on which L fixes a probability measure. We have to show that Y istrivial. Since H fixes a point in Prob(Y ) and the G-action on Prob(Y ) is stronglyproximal by Theorem 2.1, H fixes a point in Y by Lemma 3.13. So there existsy ∈ Y such that L ≤ Gy, and it follows that Gy acts minimally on X. Thereforeby Lemma 3.8 X and Y are disjoint, and since X is non-trivial and G is boundaryindivisible, this is possible only if Y is trivial. 5.3. The proof of Theorem 1.2. In this paragraph we shall give the proof ofTheorem 1.2 from the introduction. We will make use of the following result.

Proposition 5.9 (Furstenberg). Let G be a locally compact group, H ≤ G a closedsubgroup of finite covolume, and X a G-boundary. Then X is a H-boundary.


For completeness we repeat the argument from [Fur81, Prop. 4.4].

Proof. Write Q = Prob(X), and consider a closed H-invariant subspace Q′ ⊆ Q.We have to show that X ⊆ Q′. The set

X =(gH, µ) : µ ∈ g(Q′)

⊆ G/H ×Q

is a well-defined, closed, G-invariant subspace of G/H × Q. Fix a G-invariantprobability measure mG/H on G/H, and consider

Y =ν ∈ Prob(X ) : p∗

G/H(ν) = mG/H

,

where pG/H is the projection from G/H×Q onto the first factor, and p∗G/H is the in-

duced push-forward operator. Then Y is a closed (and hence compact) G-invariantsubspace of Prob(X ), and p∗

Q : Y → Prob(Q) is continuous. So p∗Q(Y ) is closed in

Prob(Q), and by strong proximality of the G-action on Q (Theorem 2.1), p∗Q(Y )

must intersect Q. Now X being the unique minimal closed G-invariant subspaceof Q, one has X ⊆ p∗

Q(Y ). For every x ∈ X, we therefore have νx ∈ Prob(X ) suchthat p∗

G/H(νx) = mG/H and p∗Q(νx) = δx. This implies mG/H gH : x ∈ g(Q′) = 1

for every x, and it easily follows that X ⊆ Q′.

Remark 5.10. In the case when H is cocompact in G, strong proximality of theaction of H on X also follows from [Gla76, II.3.1] applied to the action on Prob(X);and minimality follows [Gla76, IV.5.1] from disjointness of the G-spaces G/H andX [Gla76, III.6.1].

Theorem 5.11. Assume that H admits an amenable URS H that comes from anextremely proximal action, and such that Env(H) is co-amenable in H. Let G be alocally compact group containing H as a closed subgroup of finite covolume. Then:

(a) G is boundary indivisible.(b) More generally if L is a locally compact group such that there is a sequence

a topological group homomorphisms G = G0 → G1 → . . . → Gn = L suchthat either Gi → Gi+1 has dense image, or Gi → Gi+1 is an embeddingof Gi as a closed subgroup of finite covolume in Gi+1; then L is boundaryindivisible.

In particular whenever G maps continuously and with dense image to a productG1 ×G2, one factor Gi must be amenable.

Proof. Since H is amenable, H comes from an extremely proximal action, andEnv(H) is co-amenable in H, the group H is boundary indivisible by Proposi-tion 3.24. Now by Proposition 5.9, the property of being boundary indivisible isinherited from closed subgroups of finite covolume. Indeed if X,Y are disjoint G-boundaries, i.e. X × Y is a G-boundary, then X × Y is also a boundary for H byProposition 5.9, hence of X or Y must be trivial since H is boundary indivisible.This shows (a). Since boundary indivisibility passes to dense continuous images,and is inherited from closed subgroups of finite covolume, (b) follows from (a).

Finally if G1, G2 are as in the last statement and Xi = ∂spGi, then X1 × X2 isa boundary for G1 ×G2, which is boundary indivisible by the previous paragraph.So one factor Xi must be trivial, which exactly means that Gi is amenable byTheorem 2.2.

26 ADRIEN LE BOUDEC

Remark 5.12. In the proof of Theorem 5.11 we obtain that G is boundary indi-visible from the same property for H, which is itself deduced from Proposition 3.24(which in turn relies notably on Proposition 3.9). We note that the order in whichthe argument is developed seems to matter, in the sense that the arguments appliedto H do not seem to be applicable directly to the group G. Indeed we do not knowwhether a group G as in Theorem 5.11 falls into the setting of Proposition 3.9,i.e. we do not know whether all non-trivial G-boundaries have the same stabilizerURS. We actually believe this might be false in general.

We note at this point that the proof of Theorem 1.2 from the introduction isnow complete. Indeed the fact that a group G as in Theorem 1.2 is boundaryindivisible, as well as statement (a), is Theorem 5.11; and statement (b) followsfrom Proposition 5.8.

The following remark explains a comment from the introduction.

Remark 5.13. Theorem 1.4 provides instances of countable groups Γ being lat-tices in a group G of the form G = N o Aut(Td), such that all the assumptions ofTheorem 1.2 are satisfied (see Section 6). If M is a cocompact subgroup of Aut(Td)acting minimally on ∂Td, then L = N oM is a cocompact (hence uniformly recur-rent) subgroup of G, and L is boundary-minimal in G since ∂Td is a G-boundary.However when M is non-unimodular, then M is not co-amenable in Aut(Td), andL is not co-amenable in G. This shows that the conclusion of statement (b) inTheorem 1.2 that L is weakly co-amenable in G cannot be strengthen by sayingthat L is co-amenable in G.

5.4. The proof of Theorem 1.1. Recall that if G is a topological group, thequasi-center QZ(G) of G is the subgroup of G containing the elements g ∈ Ghaving an open centralizer. Note that QZ(G) contains the elements having a dis-crete conjugacy class, so in particular it contains all discrete normal subgroups.Recall also that the elliptic radical of G is the largest normal subgroup of G inwhich every compact subset generates a relatively compact subgroup. It is a closedcharacteristic subgroup of G.

We say that two groups G1, G2 are commensurable up to compact kernelsif there exist Ki ≤ Hi ≤ Gi such that Hi is open and of finite index in Gi, Ki is acompact normal subgroup of Hi, and H1/K1 and H2/K2 are isomorphic.

The following is slightly more complete than Theorem 1.1 from the introduction.

Theorem 5.14. Let Γ be a countable group whose Furstenberg URS AΓ comes froma faithful and extremely proximal action, and assume that Env(AΓ) is co-amenablein Γ. Let G be a locally compact group containing Γ as a lattice, and H a groupcommenurable with G up to compact kernels. Consider the following properties:

(a) Env(AΓ) is finitely generated;(b) Env(AΓ) has finite index in Γ, and Γ admits a finitely generated subgroup

with finite centralizer;(c) Env(AΓ) has finite index in Γ and Env(AΓ) has finite abelianization.

Then:

• (a), (b) both imply that H cannot be a product of two non-compact groups.


• (c) implies that any continuous morphism with dense image from H to aproduct of locally compact groups H → G1 ×G2 is such that one factor Gi

is compact.

Proof. For simplicity we give the proof for H = G. The general case follows thesame lines.

Of course we may assume that Env(AΓ) is non-trivial, since otherwise there isnothing to prove. According to Proposition 4.6 we have in particular that Γ ismonolithic, and Mon(Γ) = [Env(AΓ),Env(AΓ)]. For simplicity in all the proof wewrite E = Env(AΓ) and M = Mon(Γ) = [E,E].

Assume that φ : G → G1 × G2 is continuous with dense image, and denote bypi the projection G1 × G2 → Gi, i = 1, 2. We will show that one factor must becompact. Upon modding out by the maximal compact normal subgroup of theidentity component G0, which intersects Γ trivially since Γ has no non-trivial finitenormal subgroup (Proposition 4.6), we may also assume that G0 has no non-trivialcompact normal subgroup. This implies in particular that G0 is a connected Liegroup [MZ55].

By the assumption that E is co-amenable in Γ, we can apply Theorem 5.11,which says that one factor, say G2, must be amenable. We then apply Corollary4.13, which tells us that the map p2 φ is not injective in restriction to Γ. Bydefinition of M we deduce that M ≤ φ−1(G1 × 1).

Assume now that (c) holds. Then M , being of finite index in Γ, is a lattice in G,and is contained in the closed normal subgroup φ−1(G1 × 1). Therefore we deducethat φ−1(G1 × 1) is cocompact in G, and that p2 φ(G) is a compact subgroup ofG2. Since p2 φ(G) is also dense in G2, we have that G2 is compact.

We now have to deal with (a), (b), in which case φ is the identity and G =G1 ×G2. Without loss of generality, we may assume that the projections pi(Γ) aredense. The proofs of the two cases will share a common mechanism, given by thefollowing easy fact:

Lemma 5.15. If there exists a subgroup L ≤ G whose centralizer CG(L) containsG2, CG(L) is open in G, and CG(L) ∩ Γ is finite, then G2 is compact.

Indeed, since Γ must intersect an open subgroup O ≤ G along a lattice of O, itfollows that CG(L) is compact, and a fortiori so is G2.

We start with case (a). Consider H1 = p1(E), which is normal in G1 by densityof p1(Γ). Note that H1 is compactly generated in view of the assumption that E isfinitely generated. Since M = [E,E], the group H1/M is abelian, and therefore ofthe form Zn × Rm ×C for some compact group C. It follows that the group Q1 =H1/H

01 admits a discrete cocompact normal subgroup ∆, which is an extension of

M by a free abelian group. Being characteristically simple and non-amenable, thegroup M has trivial elliptic radical, so the group ∆ also has trivial elliptic radical.Now since Q1 is compactly generated, there is a compact open normal subgroup Kof Q1 such that Ko∆ has finite index in Q1 (see e.g. [BCGM16, Lem. 4.4]), so wededuce that Q1 has a compact open elliptic radical. Since any connected group hascompact elliptic radical [MZ55], we deduce that H1 has a compact elliptic radicalR, and H1/R is discrete-by-connected. The compact group R is also normal in G,and therefore we can mod out by R and assume that R is trivial, so that H0

1 isopen in H1. Since H0

1 centralizes M , any γ ∈ E such that p1(γ) belongs to H01

28 ADRIEN LE BOUDEC

centralizes M , and therefore is trivial by Proposition 4.6. Therefore H01 is open

in H1 and intersects the dense subgroup p1(E) trivially, so it follows that H01 is

trivial, and p1(E) is a discrete subgroup of G.Observe that p1(E) is centralized by G2 and normalized by G1, and hence is

normal in G. Being a discrete normal subgroup of G, p1(E) therefore lies in thequasi-center QZ(G). Since p1(E) is finitely generated, the centralizer of p1(E) in Gis actually open in G. Moreover the subgroup Γ ∩ CG(p1(E)) is normal in Γ sinceCG(p1(E)) is normal in G, but clearly does not contain M , and hence is trivialby Proposition 4.6. Therefore we can apply Lemma 5.15 with L = p1(E), and weobtain the conclusion.

We now deal with (b). Let Z be a minimal compact Γ-space on which theΓ-action is faithful and extremely proximal and such that SΓ(Z) = AΓ. By Propo-sition 5.9 (actually an easy case of it) the action of E on Z is also minimal, andit is extremely proximal by Lemma 2.5. Moreover the associated stabilizer URSremains equal to AΓ, and is also the Furstenberg URS of E by Proposition 3.26.So E satisfies all the assumptions of case (b) of the theorem, so it is enough toprove the result under the additional assumption Γ = E. In this case we haveM = [Γ,Γ] thanks to Proposition 4.6, so it follows that p2(Γ) is abelian. By den-sity of the projection the group G2 is also abelian, and hence G2 lies in the centerof G. Therefore Γ is normalized by the dense subgroup ΓG2, and it follows that Γis normal in G. In particular Γ ≤ QZ(G), and the conclusion follows by applyingLemma 5.15 with L a f.g. subgroup of Γ such that CΓ(L) is finite.

6. Groups acting on trees

6.1. Amenable URS’s and groups acting on trees. In this paragraph T is alocally finite tree, and H acts continuously on T by isometries. The assumptionthat T is locally finite is not essential here, and the results admit appropriategeneralizations for non-locally finite trees (using the compactification from [MS04,Prop. 4.2]).

Recall that the H-action on T is minimal if there is no proper invariant subtree,and of general type if H has no finite orbit in T ∪ ∂T . The following is well-known, and essentially goes back to Tits [Tit70] (see also [PV91] and Proposition2.4 for details).

Proposition 6.1. If the action of H ≤ Aut(T ) is minimal and of general type,then the action of H on ∂T is minimal and extremely proximal.

Theorem 5.11 therefore implies the following result:

Corollary 6.2. Let H ≤ Aut(T ) be a locally compact group whose action on T iscontinuous, minimal and of general type. Assume that end stabilizers are amenable,and the envelope of SH(∂T ) is co-amenable in H. Assume H embeds as a subgroupof finite covolume in G. Then whenever G maps continuously and with dense imageto a product G1 ×G2, one factor Gi must be amenable.

The conclusion of Corollary 6.2 implies in particular that whenever H embeds inG with finite covolume, then G cannot be a product of two non-amenable groups.The following example, which is largely inspired from [CM11, Ex. II.8], shows thatthe group G can nonetheless be a product of two non-compact groups.


Example 6.3. Let k = Fp((t)) be the field of Laurent series over the finite field Fp,and let α ∈ Aut(k) be a non-trivial automorphism of k. The group L = SL(2,k)acts on a (p + 1)-regular tree, 2-transitively and with amenable stabilizers on theboundary. This action extends to a continuous action of H = L oα Z, so that Hsatisfies all the assumptions of Corollary 6.2. Nevertheless H embeds diagonally inthe product G = (L o Aut(k)) × Z as a closed subgroup of finite covolume sinceG/H is compact and H and G are unimodular.

We will need the following fact. If A is a subtree of T , by the fixator of A wemean the subgroup fixing pointwise A.

Proposition 6.4. Let Γ ≤ Aut(T ) be a countable group whose action on T isminimal and of general type, and such that end-stabilizers in Γ are amenable. ThenAΓ = SΓ(∂T ), and Env(AΓ) is the subgroup generated by fixators of half-trees.

Proof. Since the Γ-action on ∂T is extremely proximal, it is also strongly proximalby Theorem 2.3. So ∂T is a Γ-boundary with amenable stabilizers, and we deducethat AΓ = SΓ(∂T ) by Proposition 3.25.

Now according to Lemma 3.7, the subgroup Env(SΓ(∂T )) = Env(AΓ) is gener-ated by the elements γ ∈ Γ whose fixed point set in ∂T has non-empty interior.Since half-trees form a basis of the topology in ∂T , the statement follows.

Before going to the proof of Corollary 1.3, we make the following observation:

Remark 6.5. For Γ acting on T (action minimal and general type) such thatthe action on ∂T is not topologically free, virtual simplicity of Γ is equivalent toΓ0 being of finite index in Γ and Γ0 having finite abelianization, where Γ0 is thesubgroup generated by fixators of half-trees. See statement (e) of Proposition 4.6.

Proof of Corollary 1.3. In view of Proposition 6.4, the assumptions on Γ implythat the Furstenberg URS of Γ comes from a faithful extremely proximal action.The fact that end-stabilizers are all non-trivial means that the action of Γ on ∂Tis not topologically free, and by the above observation virtual simplicity of Γ isequivalent to Env(AΓ) being of finite index in Γ and with finite abelianization.The first statement of the corollary therefore follows from Theorem 5.14, case (c),and the second statement from Theorem 1.2.

6.2. Groups with prescribed local action. In the next paragraphs we willillustrate the results of the previous sections on a family of groups acting on trees,which contains instances of discrete and non-discrete groups. The purpose of thisparagraph is to recall the definition and give a brief description of known propertiesof these groups.

We will denote by Ω a set of cardinality d ≥ 3 and by Td a d-regular tree. Thevertex set and edge set of Td will be denoted respectively Vd and Ed. We fix acoloring c : Ed → Ω such that neighbouring edges have different colors. For everyg ∈ Aut(Td) and every v ∈ Vd, the action of g on the star around v gives rise toa permutation of Ω, denoted σ(g, v), and called the local permutation of g at v.These permutations satisfy the identity(1) σ(gh, v) = σ(g, hv)σ(h, v)for every g, h ∈ Aut(Td) and v ∈ Vd.

30 ADRIEN LE BOUDEC

Given a permutation group F ≤ Sym(Ω), the group U(F ) introduced by Burgerand Mozes in [BM00a] is the group of automorphisms g ∈ Aut(Td) such thatσ(g, v) ∈ F for all v. It is a closed cocompact subgroup of Aut(Td).

Definition 6.6. Given F ≤ F ′ ≤ Sym(Ω), we denote by G(F, F ′) the group ofautomorphisms g ∈ Aut(Td) such that σ(g, v) ∈ F ′ for all v and σ(g, v) ∈ F for allbut finitely many v.

That G(F, F ′) is indeed a subgroup of Aut(Td) follows from (1), and we notethat we have U(F ) ≤ G(F, F ′) ≤ U(F ′). We make the following observation forfuture reference.

Remark 6.7. As it follows from the definition, any element γ ∈ G(F, F ′) fixingan edge e can be (uniquely) written as γ = γ1γ2, where each γi belongs to G(F, F ′)and fixes one of the two half-trees defined by e.

In the sequel we always assume that F ′ preserves the F -orbits in Ω (see [LB16,Lem. 3.3] for the relevance of this property in this context). These groups satisfythe following properties (see [LB16]):

(1) The group G(F, F ′) is dense in the locally compact group U(F ′). In par-ticular G(F, Sym(Ω)) is a dense subgroup of Aut(Td).

(2) G(F, F ′) admits a locally compact group topology (defined by requiring thatthe inclusion of U(F ) is continuous and open), and the action of G(F, F ′)on Td is continuous but not proper as soon as F = F ′. Endowed with thistopology, the group G(F, F ′) is compactly generated.

(3) stabilizers of vertices and stabilizers of ends in G(F, F ′) are respectively lo-cally elliptic and (locally elliptic)-by-cyclic. In particular they are amenable.

(4) G(F, F ′) is a discrete group if and only if F acts freely on Ω. When this isso, the group G(F, F ′) is therefore a finitely generated group, and stabilizersof vertices and stabilizers of ends in G(F, F ′) are respectively locally finiteand (locally finite)-by-cyclic.

When F acts freely on Ω and F = F ′, the groups G(F, F ′) are instances ofgroups obtained from a more general construction described in [LB17, Sec. 4](more precisely, a variation of it), which provides discrete groups with a continuousFurstenberg URS (the later being the stabilizer URS associated to the action onthe boundary of the tree on which these groups act). In the particular case of thegroups G(F, F ′), the Furstenberg URS can be explicitly described, see Proposition4.28 and Corollary 4.29 in [LBMB16].

In the sequel whenever we use letters F and F ′, we will always mean that F, F ′

are permutation groups on a set Ω, that F ′ contains F and preserves the F -orbitsin Ω. Following [Tit70], we will denote by G(F, F ′)+ the subgroup of G(F, F ′) gen-erated by fixators of edges, and by G(F, F ′)∗ the subgroup of index two in G(F, F ′)preserving the bipartition of Td.

The following result, also obtained in [CRW17, Prop. 9.16], supplements simplic-ity results obtained in [LB16], where the index of the simple subgroup was foundexplicitly under appropriate assumptions on the permutation groups.

Proposition 6.8. The group G(F, F ′) has a simple subgroup of finite index if andonly if F is transitive and F ′ is generated by its point stabilizers.


Proof. These conditions are necessary by [LB16, Prop. 4.7]. Conversely, assume Ftransitive and F ′ generated by its point stabilizers. By [LB16, Prop. 4.7] again,G(F, F ′)+ has index two in G(F, F ′), so in particular it is compactly generated. IfM is the monolith of G(F, F ′), which is simple and open in G(F, F ′) by [LB16, Cor.4.9], we have to show M has finite index. According to Remark 6.7, G(F, F ′)+ isalso the subgroup generated by fixators of half-trees, and therefore by Proposition4.6 M is the commutator subgroup of G(F, F ′)+. The abelianization of G(F, F ′)+

is therefore a finitely generated abelian group, which is generated by torsion ele-ments since G(F, F ′)+ is generated by locally elliptic subgroups (fixators of edges).Therefore this abelianization is finite, and it follows that M has finite index inG(F, F ′). 6.3. Boundaries of G(F, F ′). In this paragraph we use results from the previoussections in order to study the boundaries of the discrete groups G(F, F ′). The fol-lowing result shows that several properties of the set of boundaries are governed bythe permutation groups, and that rigidity phenomena occur under mild conditionsof the permutation groups.Theorem 6.9. Assume F acts freely on Ω, F = F ′, and write Γ = G(F, F ′). Thefollowing are equivalent:

(i) The subgroup of F ′ generated by its point stabilizers has at most two orbitsin Ω.

(ii) Γ/Env(AΓ) is isomorphic to one of C2, D∞ or D∞ o C2.(iii) Env(AΓ) is co-amenable in Γ.(iv) SΓ(X) = AΓ for every non-trivial Γ-boundary X.(v) Γ is boundary indivisible.

We will need preliminary results before proving Theorem 6.9.Lemma 6.10. Assume that F acts freely on Ω. The envelope of the FurstenbergURS of G(F, F ′) is equal to G(F, F ′)+.Proof. Write Γ = G(F, F ′) and Γ+ = G(F, F ′)+. According to Proposition 6.4,Env(AΓ) is the subgroup generated by fixators of half-trees in Γ. Therefore theinclusion Env(AΓ) ≤ Γ+ is clear. The converse inclusion also holds true by Remark6.7, so equality follows.

In view of Lemma 6.10 and Proposition 3.24, we are led to consider the quotientG(F, F ′)/G(F, F ′)+, and in particular study when it is amenable. To this end, wewill denote by F ′+ the subgroup of F ′ generated by its point stabilizers, and writeD = F ′/F ′+. Since F ′+ is normal in F ′, we have an action of F ′ on the set oforbits of F ′+, which factors through a free action of D.Proposition 6.11. The group Q = G(F, F ′)∗/G(F, F ′)+ is isomorphic to the groupU(D)∗, where D = F ′/F ′+ is viewed as a permutation group acting freely on theset of orbits of F ′+ in Ω. If moreover F is transitive, one has Q = D ∗D.Proof. We let O1, . . . , Or ⊆ Ω be the orbits of F ′+, and we will freely identifythe set of orbits with the integers 1, . . . , r. For every a ∈ Ω, there is a uniquei ∈ 1, . . . , r such that a ∈ Oi, and we denote i = ia.

We view the tree Td as the Cayley graph of the free Coxeter group of rank d,namely the group defined by generators x1, . . . , xd and relators x2

j = 1 for all j.

32 ADRIEN LE BOUDEC

When adding relations of the form xa = xb whenever ia = ib (i.e. a and b are in thesame F ′+-orbit), we obtain a free Coxeter group of rank r. It has a Cayley graphthat is a regular tree of degree r, and we have a surjective map p : Td → Tr.

Two elements v, w ∈ ∗dC2 = ⟨x1, . . . , xd⟩ have the same image in ∗rC2 if andonly if one can write w = v

∏wjxajxbj

w−1j for some words wj and colors aj , bj

such that iaj = ibj. Since the inverse of wj is equal to the word obtained from wj

by reversing the order, we have:

Lemma 6.12. Two vertices v, w of Td have the same projection in Tr if and onlyif the distance between v and w is even, say d(v, w) = 2m, and if (a1, . . . , a2m) isthe sequence of colors from v to w, then the word (ia1 , . . . , ia2m) is a concatenationof palindromes of even length.

Lemma 6.13. For every g ∈ G(F, F ′) and every vertex v on Td, the image ofσ(g, v) in D = F ′/F ′+ does not depend on v. We denote by σg ∈ D the corre-sponding element, which is trivial when g ∈ G(F, F ′)+.

Proof. If v, w are adjacent vertices and a is the color of the edge between them,then σ(g, v)(a) = σ(g, w)(a). So σ(g, v)σ(g, w)−1 ∈ F ′+. The first statementfollows by connectedness. The fact that σg is trivial on G(F, F ′)+ is then clearbecause g 7→ σg is a morphism according to the first statement, which vanishes onfixators of edges.

Note that the set of edges of Tr inherits a natural coloring by the integers 1, . . . , r.

Lemma 6.14. There is a natural morphism φ : G(F, F ′) → Aut(Tr) such thatker(φ) = G(F, F ′)+ and Im(φ) = U(D).

Proof. We shall first define an action of G(F, F ′) on the set of vertices of Tr. Letg ∈ G(F, F ′). Let v, w be two vertices of Td, and (a1, . . . , an) be the sequence ofcolors from v to w. If v = v1, . . . , vn+1 = w are the vertices between v and w,then the sequence of colors from g(v) to g(w) is (σ(g, v1)(a1), . . . , σ(g, vn)(an)). Ifσg is the element defined in Lemma 6.13, then one has iσ(g,vj)(aj) = σg(iaj ) for allj = 1, . . . , n. This shows in particular that if v, w satisfy the condition of Lemma6.12, then the same holds for g(v) and g(w). This means that for every vertex xof Tr, the formula(2) φ(g)(x) := p(gx),where x is any vertex of Td such that p(x) = x, is a well-defined action of G(F, F ′)on Tr. The fact that the tree structure is preserved is clear. Note that for everyg ∈ G(F, F ′), all the local permutations of φ(g) are equal to σg: for every vertex xof Tr, one has σ(φ(g), x) = σg. In particular the image of φ lies inside U(D).

We shall prove that ker(φ) = G(F, F ′)+. Let g ∈ G(F, F ′) fixing an edge in Td.Then φ(g) also fixes an edge of Tr by (2). Moreover one has σg = 1 (Lemma 6.13),and it follows from the previous paragraph that all local permutations of φ(g) aretrivial. This implies g ∈ ker(φ). Conversely, we let g be an element of ker(φ),and we prove that g ∈ G(F, F ′)+. Note that since φ(g) is trivial, one has σg = 1,i.e. all local permutations of g are in F ′+. Now let v be any vertex. By Lemma6.12, the sequence of colors (a1, . . . , a2n) from v to g(v) gives rise to a sequence(ia1 , . . . , ia2n) that is a concatenation of palindromes. For simplicity we treat thecase where (ia1 , . . . , ia2n) is a palindrome; the general case consists in repeating the


argument for this case. Let v = v1, . . . , v2n+1 = g(v) be the vertices between vand g(v) (note that vn is the midpoint between v and g(v)). Since (ia1 , . . . , ia2n)is a palindrome, one easily checks that there are elements g1, . . . , gn such that gj

belongs to the stabilizer of vj in G(F, F ′+) and g′ = g1 . . . gng fixes the vertex v(this is obtained by successively folding the geodesic [v, g(v)] onto itself startingfrom its midpoint in order to bring back g(v) to v with g1 . . . gn). We now invokethe following easy fact, whose verification is left to the reader.

Lemma 6.15. Let γ ∈ G(F, F ′) fixing a vertex w, and such that σ(γ,w) ∈ F ′+.Then γ ∈ G(F, F ′)+.

We apply Lemma 6.15 to g′ and all gj ’s, and deduce that g = g−1n . . . g−1

1 g′

belongs to G(F, F ′)+ as desired.The last thing that remains to be proved in the statement of Lemma 6.14 is that

the image of φ is equal to U(D). The fact that φ(g) always belongs to U(D) hasalready been observed. For the converse inclusion, observe that since G(F, F ′) actstransitively on the vertices of Tr (as it is already the case on Td), it is enough tocheck that the image of φ contains U(D)x for some vertex x of Tr. Now since Dacts freely on 1, . . . , r, the map U(D)x → D, γ 7→ σ(γ, x), is an isomorphism.Therefore it is enough to see that any action on the star around x on Tr canrealized by an element of G(F, F ′), and this is indeed the case (see e.g. Lemma 3.4in [LB16]).

To finish the proof of the proposition, remark that the image of G(F, F ′)∗ by φis precisely U(D)∗. When F is transitive then D is also transitive, so that U(D)∗

has two orbits of vertices and one orbit of edges, and therefore splits as the freeproduct D ∗D. Remark 6.16. The case F = F ′ is allowed in Proposition 6.11, so that the con-clusion also holds for the groups U(F ) from [BM00a].

Proposition 6.11 naturally leads us to isolate the following three situations. Wekeep the previous notation, so that r is the number of orbits of F ′+ in Ω, D =F ′/F ′+ and Q = G(F, F ′)∗/G(F, F ′)+:(1) r = 1. In this case Tr is a segment of length one, and D and Q are trivial.(2) r = 2. Tr is a bi-infinite line, and this case splits into two disjoint sub-cases:

(a) If F ′ is intransitive then D is trivial, and Q = U(1)∗ = Z (generated by atranslation of Tr of length 2).

(b) If F ′ is transitive then we have D = Sym(2) and Q = D ∗D = D∞.(3) r ≥ 3. Then Q = U(D)∗ is a virtually free group (since it acts vertex transi-

tively and with trivial edge stabilizers of Tr).Theorem 6.9 says that all properties stated there hold true if and only if r ∈

1, 2. A sufficient condition for having r = 1 is for instance that F acts transitivelyon Ω, F = F ′ and F ′ acts primitively, or quasi-primitively on Ω. Recall thata permutation group is quasi-primitive if every non-trivial normal subgroup actstransitively.

But Theorem 6.9 also applies beyond the case of quasi-primitive permutationgroups. For example a situation giving rise to case (2) (a) is when F ′ has a fixedpoint and acts transitively on the complement. Examples giving rise to case (2) (b)are for instance obtained by taking F ′ = Sym(n)≀C2 = Sym(n)≀⟨τ⟩ acting naturally

34 ADRIEN LE BOUDEC

on 2n letters, and F the subgroup generated by ((cn, cn), 1) and ((1, 1), τ), wherecn is a cycle of order n.

Proof of Theorem 6.9. (i) ⇒ (ii) follows from Lemma 6.10 and Proposition 6.11(and the discussion following its proof). (ii) ⇒ (iii) is clear. (iii) ⇒ (iv) is Proposi-tion 3.24. (iv) ⇒ (v) is guaranteed by Proposition 3.9 and the fact that AΓ is nota point. Finally assume that (i) does not hold, i.e. F ′+ has at least three orbitsin Ω, and write Γ+ = G(F, F ′)+. By Proposition 6.11, the group Q = Γ/Γ+ hasa subgroup of finite index that is free of rank at least 2. So there exist non-trivialQ-boundaries, and a fortiori these are non-trivial Γ-boundaries. If X is such aboundary, then Γ+ acts trivially on X. Since Γ+ also acts minimally on ∂Td, itfollows that X and ∂Td are disjoint Γ-boundaries, contradicting (v). Thereforeproperty (v) implies property (i), and the proof is complete.

6.4. Weakly co-amenable subgroups. In this paragraph we show that sub-groups of the groups G(F, F ′) satisfy the following dichotomy:

Proposition 6.17. Assume that F acts freely transitively and F ′ acts primitivelyon Ω. Then any subgroup of G(F, F ′) is either (locally finite)-by-cyclic (and henceamenable) or weakly co-amenable.

We will need the following lemma.

Lemma 6.18. Assume that F ′ acts primitively on Ω, and take two subgroupsH1 = H2 in the Furstenberg URS of G(F, F ′). Then ⟨H1,H2⟩ = G(F, F ′)∗.

Proof. Write Γ = G(F, F ′). Recall from [LBMB16, Prop. 4.28] that the FurstenbergURS of Γ consists of subgroups Γ0

ξ , ξ ∈ ∂Td, where Γ0ξ is the set of elements acting

trivially on a neighbourhood of ξ. Given ξ = η ∈ ∂Td, we show that the subgroupΛ generated by Γ0

ξ and Γ0η must be equal to G(F, F ′)∗.

Take a vertex v on the geodesic from ξ to η, let e1, e2 be the edges containingv and pointing towards ξ and η, and a, b the colors of e1, e2. Denote by K(v) thesubgroup of Γ consisting of elements γ fixing v and such that σ(γ,w) ∈ F for everyw = v. Since F ′ is primitive, F ′ is generated by the point stabilizers F ′

a and F ′b.

This implies that every element of K(v) may be written as a product of elementsfixing either the half-tree defined by e1 containing ξ, or the half-tree defined by e2containing η, so that K(v) ≤ Λ. Since v was arbitrary, we also have K(v′) ≤ Λ forv′ a neighbour of v on the geodesic [ξ, η]. The conclusion now follows since for twoneighbouring vertices v, v′, the subgroups K(v),K(v′) always generate G(F, F ′)∗

[LB16, Cor. 3.10]. Proof of Proposition 6.17. Write Γ = G(F, F ′), and let Λ be a subgroup of Γ that isnon-amenable, equivalently whose action of Td is of general type. By Proposition5.7 we have to show that Λ fixes no probability measure on any non-trivial Γ-boundary. Argue by contradiction and assume that X is a non-trivial Γ-boundaryon which Λ fixes a probability measure µ. According to Theorem 6.9, we haveSΓ(X) = AΓ. Therefore by Proposition 3.2 there exist an almost 1-1 extensionη : X → X and a factor map π : X → AΓ.

Let Q ⊂ Prob(X) be the set of ν such that η∗ν = µ, and write R = π∗(Q),which is a closed Λ-invariant subset of Prob(AΓ). Since the action of Λ on ∂Td

is strongly proximal and since AΓ is a factor of ∂Td [LBMB16, Prop. 2.10-4.28],


we deduce that R contains some Dirac measures. Let H ∈ AΓ such that there isν ∈ Prob(X) with η∗ν = µ and π∗ν = δH . Such a measure ν must be supportedin the set of (x,H) ∈ X, and it follows that µ is supported in the set of H-fixedpoints in X (because (x,H) ∈ X implies that H ≤ Gx by upper semi-continuityof the stabilizer map). But since Λ does not fix any point in AΓ, we may findanother H ′ ∈ AΓ such that δH′ ∈ R, so that the same argument shows that H ′ alsoacts trivially on the support of µ. By Lemma 6.18 the subgroups H,H ′ generateG(F, F ′)∗, which is of index two in Γ. Therefore any point in the support of µ hasa Γ-orbit of cardinality at most two, which is absurd since Γ acts minimally on Xand X is non-trivial by assumption. Remark 6.19. Assume F acts freely transitively and F ′ acts primitively, andwrite Γ = G(F, F ′). Let Λ ≤ Γ such that the Λ-action on Td is of general type butwith a proper Λ-invariant subtree. For instance one could take for Λ the subgroupgenerated by two hyperbolic elements with sufficiently far apart axis. Then Λ isnot co-amenable in Γ (see the argument in the proof of Theorem 2.4 in [CM09]),but Λ is weakly co-amenable in Γ by Proposition 6.17.Remark 6.20. We mention that when F ′ is primitive, following the proof of Corol-lary 4.14 from [LBMB16] (with minor modifications), one could prove that everynon-trivial G(F, F ′)-boundary factors onto ∂Td. This would provide an alternativeproof of Proposition 6.17.6.5. Lattice embeddings of the groups G(F, F ′). In this section we study howthe discrete groups G(F, F ′) can embed as lattices in some locally compact groups.The purpose of this paragraph is twofold:(1) First we apply previous results of the article to the family of groups G(F, F ′)

and deduce some properties of general locally compact groups containing agroup G(F, F ′) as a lattice (Corollary 6.22).

(2) Second we explain how the groups G(F, F ′) embed as lattices in some locallycompact wreath products. This will be the content of §6.5.2 below.

Remark 6.21. Maybe it is worth pointing out that instances of lattice embeddingsof the groups G(F, F ′) already appeared in [LB16]. Indeed under appropriateassumptions on permutation groups F ≤ F ′,H ≤ H ′, the inclusion of G(F, F ′) inG(H,H ′) has discrete and cocompact image [LB16, Cor. 7.4].Corollary 6.22. Assume that F acts freely transitively on Ω, and that F ′ is gen-erated by its point stabilizers. Let G be a locally compact group containing G(F, F ′)as a lattice. Then the conclusions of Corollary 1.3 hold.Proof. The assumptions on F, F ′ imply that G(F, F ′) is virtually simple by Propo-sition 6.8, so Corollary 1.3 applies. Remark 6.23. In the setting of Corollary 6.22, although G(F, F ′) cannot be alattice in a product, it happens that there exist non-discrete groups G1, G2 suchthat G(F, F ′) embeds as a discrete subgroup of G1 × G2 with injective and denseprojection to each factor. For instance if F1, F2 are permutation groups such thatF Fi ≤ F ′ and we set Gi = G(Fi, F

′), then the diagonal embedding of G(F, F ′)in G1 × G2 has this property as soon as F1 ∩ F2 = F . See [LB16, Lem. 3.4 and§7.1].

36 ADRIEN LE BOUDEC

6.5.1. Locally compact wreath products. In this paragraph we introduce some ter-minology that will be used in the sequel.

Let Ω be a set, B a group and A a subgroup of B. We will denote by BΩ,A theset of functions f : Ω → B such that f(x) ∈ A for all but finitely many x ∈ Ω.Note that BΩ,A is a group.

Definition 6.24 ([Cor17]). If H is a group acting on Ω, the semi-restrictedpermutational wreath product B ≀AΩ H is the semi-direct product BΩ,A o H,where h ∈ H acts on f ∈ BX,A by (hf)(x) = f(h−1x).

The extreme situations when A = 1 and when A = B correspond respectivelyto the restricted and the unrestricted wreath product. When A = 1, we shall writeB ≀Ω H for the restricted wreath product. Also for simplicity we will sometimessay “wreath product” B ≀AΩ H instead of “semi-restricted permutational wreathproduct”.

When A is a compact group and H a locally compact group acting continuouslyon Ω, the group AΩ o H is a locally compact group for the product topology. Ifmoreover B is locally compact and A is compact open in B, there is a naturallocally compact group topology on B ≀AΩ H, defined by requiring that the inclusionof AΩ oH in B ≀AΩ H is continuous and open. See [Cor17, Sec. 2].

In the remaining of the article we shall be interested in the study of certainlattices in some locally compact groups B ≀AΩ H. A few remarks are in order:

Lemma 6.25. Let A,B,Ω and H as above.(a) Assume Γ1 is a lattice in BΩ,A and Γ2 is a lattice in H that normalizes Γ1.

Then Γ1 o Γ2 is a lattice in B ≀AΩ H.(b) For B ≀AΩ H to contain a lattice, it is necessary that H contains a lattice.

Proof. For the first statement, see [Rag72, Lem. I.1.6-7]. For the second statement,observe that if Γ ≤ B ≀AΩ H is a lattice, the intersection ΓA = Γ ∩ (AΩ o H) is alattice in AΩoH since AΩoH is open in B ≀AΩH. The subgroup AΩ being compact,the projection of ΓA to H is discrete, and hence is a lattice in H.

Recall that there are various notions of irreducibility for a lattice in a directproduct of groups. In general whether all these notions coincide depends on thecontext. We refer to [CM12, 2.B] and [CM09, 4.A] for detailed discussions. In thesetting of wreath products, we will use the following terminology:

Definition 6.26. A lattice Γ in B ≀AΩ H is an irreducible lattice if Γ has anon-discrete projection to the group H.

This definition implies that neither Γ nor its finite index subgroups can be ofthe form Γ1 o Γ2 as in Lemma 6.25.

Lemma 6.27. If the group BΩ,A does not contain any lattice, then any lattice inB ≀AΩ H is irreducible.

Proof. If Γ ≤ B≀AΩH is a lattice with a discrete projection Λ to H, then the subgroupB ≀AΩ Λ contains Γ as a lattice, and it follows that Γ intersects the subgroup BΩ,A,which is open in B ≀AΩ Λ, along a lattice of BΩ,A.


Remark 6.28. Of course if B admits no lattice then the same holds for BΩ,A.More interestingly, there are finite groups A,B for which BΩ,A fails to admit anylattice (provided Ω is infinite). This is for instance the case when A = B and anynon-trivial element of B has a non-trivial power in A. Consequently all lattices inB ≀AΩ H are irreducible by Lemma 6.27 (for arbitrary H).

Proof of Remark 6.28. We claim that the above condition on A,B actually impliesthat BΩ,A has no infinite discrete subgroup. For every finite Σ ⊂ Ω, we writeOΣ ≤ BΩ,A for the subgroup vanishing on Σ, and UΣ = OΣ ∩ AΩ. Assume Γ is adiscrete subgroup of BΩ,A, so that there is a finite Σ ⊂ Ω such that Γ ∩ UΣ = 1.The assumption on A,B is easily seen to imply that any non-trivial subgroup ofOΣ intersects UΣ non-trivially. Therefore Γ ∩OΣ = 1, and OΣ being of finite indexin BΩ,A, Γ is finite.

It should be noted that the existence of an irreducible lattice in B ≀AΩ H forcesH to be non-discrete and B to be non-trivial. However this does not force BΩ,A tobe non-discrete, and as we will see below, interesting examples already arise whenB is finite and A is trivial.

6.5.2. The proof of Theorem 1.4. Let n ≥ 2 and d ≥ 3. We denote by Σn theset of integers 0, . . . , n− 1, and by Σ(Vd)

n the set of functions f : Vd → Σn withfinite support, where the support of f is the set of v such that f(v) = 0. We willalso write fv for the image of v by f , and sometimes use the notation (fv) for thefunction f .

We consider the graph Xn,d whose set of vertices is the set of pairs (f, e), wheref belongs to Σ(Vd)

n and e ∈ Ed, and edges emanating from a vertex (f, e) are of twotypes:

• type 1: (f, e′) is connected to (f, e) if e′ ∈ Ed is a neighbour of e (i.e. if eand e′ share exactly one vertex);

• type 2: (f ′, e) is connected to (f, e) if the function f ′ is obtained from f bychanging the value at exactly one vertex of e.

Note that since any e ∈ Ed has 2(d − 1) neighbours and Σn has cardinality n,every vertex of Xn,d has 2(d− 1) neighbours of type 1 and 2(n− 1) neighbours oftype 2. The graph Xn,d is almost the wreath product of the complete graph on nvertices with the tree Td, see below.

Let Sn be the group of permutations of Σn. For σ ∈ Sn and i ∈ Σn, we will writeσ · i the action of σ on i. The stabilizer of 0 ∈ Σn in Sn is obviously isomorphicto Sn−1, and by abuse of notation we will denote it Sn−1. In particular whenviewing Sn−1 as a subgroup of Sn, we will always implicitly mean that Sn−1 isthe subgroup of Sn acting only on 1, . . . , n− 1.

Definition 6.29. We will denote by Gn,d the wreath product Sn ≀Sn−1Vd

Aut(Td).

Groups of the form Gn,d were considered in [Cor17, Ex. 2.6]. We will denoteUn,d = S

Vd,Sn−1n , so that Gn,d = Un,d oAut(Td). We endow Gn,d with the topology

such that sets of the form ((σv)U1, γU2) form a basis of neighbourhoods of ((σv), γ),where U1 and U2 belong to a basis of the identity respectively in SVd

n−1 and inAut(Td). This defines a totally disconnected locally compact group topology onGn,d (see [Cor17, Prop. 2.3]). We note that the case n = 2 is somehow particular,

38 ADRIEN LE BOUDEC

as U2,d is a discrete subgroup of G2,d, and G2,d is just the restricted wreath productG2,d = C2 ≀Vd

Aut(Td) = (⊕VdC2) o Aut(Td).

Proposition 6.30. The group Gn,d acts by automorphisms on the graph Xn,d bypreserving the types of edges. Moreover the action is faithful, continuous, properand transitive on the set of vertices.

Proof. The group Gn,d is a subgroup of the unrestricted permutational wreathproduct of Sn and Aut(Td). The latter group has a faithful action on the set offunctions Vd → Σn, given by

((σv), γ) · (fv) = (σv · fγ−1v).

The group Gn,d preserves Σ(Vd)n because σv fixes 0 almost surely if (σv) belongs to

Un,d. Now the projection from Gn,d onto Aut(Td) induces an action of Gn,d on theset Ed, and we will consider the diagonal action of Gn,d on Σ(Vd)

n × Ed. In otherwords, if g = ((σv), γ) ∈ Gn,d, and ((fv), e) ∈ Xn,d,

(3) g · ((fv), e) =((σv · fγ−1v), γe

).

Fix x = ((fv), e) ∈ Xn,d and g = ((σv), γ) ∈ Gn,d, and let x′ be a neighbour ofx. If x′ is of type 1, then we have x′ = ((fv), e′), where e and e′ share a vertex win Td. Then γe and γe′ have the vertex γw in common, so that by the formula (3),g · x′ is a neighbour of type 1 of g · x in Xn,d. Now if x′ is of type 2, then we maywrite x′ = ((f ′

v), e) with f ′v = fv if and only if v = w, where w is one of the two

vertices of e. It follows that σv · fγ−1v = σv · f ′γ−1v if and only if v = γw, so that by

(3) g · x′ is a neighbour of type 2 of g · x. This shows that the action is by graphautomorphisms and preserves the types of edges.

Lemma 6.31. Let e ∈ Ed, and let K be the stabilizer of e in Aut(Td). Then thestabilizer of the vertex ((0), e) in Gn,d is the compact open subgroup

∏Sn−1 oK.

Proof. That g = ((σv), γ) fixes ((0), e) exactly means by (3) that σv fixes 0 for allv and that γ fixes e.

So the fact that the action is continuous and proper follows from the lemma,and the transitivity on the set of vertices is an easy verification.

Consider now the free product Cd ∗Cd of two cyclic groups of order d, acting onits Bass-Serre tree Td with one orbit of edges and two orbits of vertices. Denote byCn the cyclic subgroup of Sn generated by the cycle (0, . . . , n− 1), and set

Γn,d := Cn ≀Vd(Cd ∗ Cd) ≤ Gn,d.

Remark that Cd ∗ Cd has a split morphism onto Cd, whose kernel acts on Td

with two orbits of vertices and is free of rank d − 1. Therefore Γn,d splits asΓn,d = (C2

n ≀ Fd−1) o Cd.

Lemma 6.32. Γn,d ≤ Gn,d acts freely transitively on the vertices of Xn,d.

Proof. This is clear: the image of the vertex ((0), e) by an element ((σv), γ) is((σv · 0), γe), so both transitivity and freeness follow from the fact that the actionsof Cn on Σn and of Cd ∗ Cd on Ed have these properties.


We now explain how the groups G(F, F ′) act on the graphs Xn,d. In the sequelF, F ′ denote two permutation groups on Ω such that F ≤ F ′ and F ′ preserves theorbits of F , and we denote by n the index of F in F ′.

Fix a bijection between Σn = 0, . . . , n− 1 and F ′/F , such that 0 is sent to theclass F . The action of F ′ on the coset space F ′/F induces a group homomorphism

α : F ′ → Sn,

such that α(F ) lies inside Sn−1. For γ ∈ G(F, F ′) and v ∈ Vd, writeργ,v = α(σ(γ, γ−1v)) ∈ Sn.

Note that ργ,v ∈ Sn−1 if and only if σ(γ, γ−1v) ∈ F .

Proposition 6.33. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′. Themap φ : G(F, F ′) → Gn,d, γ 7→ φ(γ) = ((ργ,v), γ) is a well-defined, injective groupmorphism with closed and cocompact image.

Proof. The map φ is well-defined because ργ,v ∈ Sn−1 for all but finitely many v,and injectivity is clear. The fact that φ is a morphism follows from the cocycleidentity (1) satisfied by local permutations. The intersection between Im(φ) andthe open subgroup SVd

n−1 oAut(Td) is φ(U(F )). The latter is a closed subgroup ofGn,d since U(F ) is closed in Aut(Td), so Im(φ) is closed. The fact that Im(φ) iscocompact will follow from Proposition 6.30 and Proposition 6.34 below.

In the sequel for simplicity we will also write G(F, F ′) for the image ofφ : G(F, F ′) → Gn,d, γ 7→ φ(γ) = ((ργ,v), γ) .

In particular when speaking about an action of G(F, F ′) on the graph Xn,d, wewill always refer to the action defined in Proposition 6.30, restricted to G(F, F ′).This means that γ ∈ G(F, F ′) acts on (f, e) ∈ Xn,d by

γ · (f, e) = (fγ , γe),where(4) (fγ)v = ργ,v · fγ−1v = α(σ(γ, γ−1v)) · fγ−1v.

This action should not be confused with the standard action ((fv), e) 7→ ((fγ−1v), γe)coming from the inclusion of G(F, F ′) in Aut(Td).

Proposition 6.34. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′.(a) The group G(F, F ′)∗ acts cocompactly on Xn,d. When F is transitive on Ω,

the group G(F, F ′)∗ acts transitively on vertices of Xn,d.(b) The stabilizer of a vertex ((0), e) ∈ Xn,d in G(F, F ′) is the stabilizer of e in

U(F ). In particular the action of G(F, F ′) on Xn,d is proper.Therefore when F acts freely transitively on Ω, the group G(F, F ′)∗ acts freelytransitively on the vertices of Xn,d.

Proof. We show that for every vertex x = ((fv), e) of Xn,d, there is g ∈ G(F, F ′)∗

such that g · x = ((0), e). Since U(F ) preserves the vertices of this form, and sincethe number of orbits of U(F )∗ ≤ G(F, F ′)∗ on Ed is finite and is equal to one whenF is transitive [BM00a], statement (a) will follow.

We argue by induction on the cardinality N of the support of (fv). There isnothing to show if N = 0. Assume N ≥ 1, and let v0 ∈ Vd with fv0 = 0 and such

40 ADRIEN LE BOUDEC

that v0 maximizes the distance from e among vertices v such that fv = 0. Let e0be the edge emanating from v0 toward e (if v0 belongs to e then e0 = e), and leta ∈ Ω be the color of e0. We also denote by T 1 and T 2 the two half-trees definedby e0, where T 1 contains v0. For every b ∈ Ω, b = a, we denote by e0,b the edgecontaining v0 and having color c(e0,b) = b, and by T 1,b the half-tree defined by e0,b

not containing v0.By assumption the permutation group F ′ preserves the F -orbits in Ω, so we have

F ′ = FF ′a. The subgroup α(F ′) ≤ Sn being transitive, it follows from the previous

decomposition that there exists σ ∈ F ′a such that α(σ) · fv0 = 0. For every b = a,

we choose σb ∈ F such that σb(b) = σ(b), and we consider the unique elementh ∈ Aut(Td) whose local permutations are σ(h, v) = 1 if v ∈ T 2; σ(h, v0) = σ;and σ(h, v) = σb for every v ∈ T 1,b and every b = a. It is an easy verification tocheck that h is a well-defined automorphism of Td, and h ∈ G(F, F ′) because allbut possibly one local permutations of h are in F .

Note that h fixes e by construction. Write h(x) = ((ϕv), e). We claim that thesupport of (ϕv) has cardinality N − 1. Since h fixes v0, by (4) we have ϕv0 =α(σ(h, v0)) · fv0 = α(σ) · fv0 = 0. Moreover we also have ϕv = fv for every v in T 2

because h acts trivially on T 2. Finally by the choice of v0 we had fv = 0 for everyv = v0 in T 1, and since σ(h, v) ∈ F for all these v and α(F ) fixes 0, we still haveϕv = 0 for every v in T 1, v = v0. This proves the claim, and the conclusion followsby induction.

Statement (b) follows from Lemma 6.31, and the last statement follows from (a)and (b) and the fact that U(F ) acts freely on Ed when F acts freely on Ω.

Propositions 6.33-6.34 and Lemma 6.32 imply Theorem 1.4 from the introduc-tion. Note that when F acts freely transitively on Ω, we have an explicit descriptionof a generating subset of the groupG(F, F ′)∗ whose associated Cayley graph isXn,d.For, fix an edge e0 ∈ Ed, whose color is a ∈ Ω and whose vertices are v0, v1, anddenote x0 = ((0), e0) ∈ Xn,d. For i = 0, 1, let Si be the set of γ ∈ G(F, F ′) fixing vi

and such that σ(γ, v) ∈ F for every v = vi, and σ(γ, vi) is non-trivial and belongsto F ∪ F ′

a. Then S = S0 ∪ S1 generates G(F, F ′)∗, and Cay(G(F, F ′)∗, S) → Xn,d,γ 7→ γx0, is a graph isomorphism. Moreover neighbours of type 1 (resp. type 2)of a vertex of Xn,d are labeled by elements s ∈ Si such that σ(γ, vi) ∈ F (resp.σ(γ, vi) ∈ F ′

a).

We end the article by observing that there are possible variations in the definitionof the graph Xn,d. If Kn is the complete graph on n vertices, let Kn ≀ Td be thewreath product of the graphs Kn and Td (sometimes also called the lamplightergraph over Td): the vertex set is Σ(Vd)

n × Vd, and there is an edge between (f, v)and (f ′, v′) if and only if either f = f ′ and v, v′ are adjacent in Td, or v = v′ andf(w) = f ′(w) if and only if w = v. Again if n is the index of F in F ′, the groupG(F, F ′) acts on Kn ≀ Td by γ · (f, v) = (fγ , γv), where fγ is given by (4). Theprevious arguments for the graph Xn,d carry over to this graph, so that we have:

Proposition 6.35. Let F ≤ F ′ ≤ Sym(Ω), and n the index of F in F ′. ThenG(F, F ′) acts properly and cocompactly on the graph Kn ≀ Td.

The reason why we considered the graph Xn,d instead of Kn ≀ Td is to obtain,under the assumption that F acts freely on Ω, a free action of G(F, F ′)∗ on the


set of vertices. In the case of Kn ≀ Td, the stabilizer of a vertex in G(F, F ′)∗ isfinite, but non-trivial. We note that it might be interesting to investigate whetherthe generalized wreath products of graphs from [Ers06] could provide other kindof interesting groups of automorphisms.

Yet another possibility is to take the same vertex set as Xn,d, but declaring thatthere is an edge between (f, e) and (f ′, e′) if e = e′ share a vertex w and fv = f ′

v

for every v = w. This graph Zn,d has larger degree, namely 2(d − 1)n. Again allthe results proved above for Xn,d remain true. In the case d = 2, one may checkthat Zn,2 is the Diestel-Leader graph DL(n, n), so that Zn,d may be thought of as“higher dimensional” versions of these graphs.

References[BCGM16] U. Bader, P.E. Caprace, T. Gelander, and S. Mozes, Lattices in amenable groups,

arXiv:1612.06220 (2016).[BF14] U. Bader and A. Furman, Boundaries, rigidity of representations, and Lyapunov ex-

ponents, Proceedings of ICM (2014).[BFS15] U. Bader, A. Furman, and R. Sauer, On the structure and arithmeticity of lattice

envelopes, C. R. Math. Acad. Sci. Paris 353 (2015), no. 5, 409–413.[BKKO14] E. Breuillard, M. Kalantar, M. Kennedy, and N. Ozawa, C∗-simplicity and the unique

trace property for discrete groups, arXiv:1410.2518v2 (2014).[BM97] M. Burger and Sh. Mozes, Finitely presented simple groups and products of trees, C.

R. Acad. Sci. Paris Sér. I Math. 324 (1997), no. 7, 747–752.[BM00a] , Groups acting on trees: from local to global structure, Inst. Hautes Études

Sci. Publ. Math. (2000), no. 92, 113–150 (2001).[BM00b] , Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000),

no. 92, 151–194.[BM02] M. Burger and N. Monod, Continuous bounded cohomology and applications to rigidity

theory, Geom. Funct. Anal. 12 (2002), no. 2, 219–280.[BNW08] L. Bartholdi, M. Neuhauser, and W. Woess, Horocyclic products of trees, J. Eur. Math.

Soc. (JEMS) 10 (2008), no. 3, 771–816.[BQ14] Y. Benoist and J-F. Quint, Lattices in S-adic Lie groups, J. Lie Theory 24 (2014),

no. 1, 179–197.[BS06] U. Bader and Y. Shalom, Factor and normal subgroup theorems for lattices in products

of groups, Invent. Math. 163 (2006), no. 2, 415–454.[Cap16] P-E. Caprace, Non-discrete simple locally compact groups, to appear in the Proceedings

of the 7th European Congress of Mathematics (2016).[CFK12] Y. Cornulier, D. Fisher, and N. Kashyap, Cross-wired lamplighter groups, New York

J. Math. 18 (2012), 667–677.[CM09] P-E. Caprace and N. Monod, Isometry groups of non-positively curved spaces: discrete

subgroups, J. Topol. 2 (2009), no. 4, 701–746.[CM11] , Decomposing locally compact groups into simple pieces, Math. Proc. Cam-

bridge Philos. Soc. 150 (2011), no. 1, 97–128.[CM12] , A lattice in more than two Kac-Moody groups is arithmetic, Israel J. Math.

190 (2012), 413–444.[CM14] , Relative amenability, Groups Geom. Dyn. 8 (2014), no. 3, 747–774.[Cor17] Y. Cornulier, Locally compact wreath products, arXiv:1703.08880 (2017).[CR09] P-E. Caprace and B. Rémy, Simplicity and superrigidity of twin building lattices, In-

vent. Math. 176 (2009), no. 1, 169–221.[CRW17] P-E. Caprace, C. Reid, and Ph. Wesolek, Approximating simple locally compact groups

by their dense locally compact subgroups, arXiv:1706.07317v1 (2017).[DGO11] F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotat-

ing families in groups acting on hyperbolic spaces, arXiv:1111.7048 (2011).

42 ADRIEN LE BOUDEC

[DM16] B. Duchesne and N. Monod, Group actions on dendrites and curves,arXiv:1609.00303v3 (2016).

[Dym15] T. Dymarz, Envelopes of certain solvable groups, Comment. Math. Helv. 90 (2015),no. 1, 195–224.

[Dyu00] A. Dyubina, Instability of the virtual solvability and the property of being virtuallytorsion-free for quasi-isometric groups, Internat. Math. Res. Notices (2000), no. 21,1097–1101.

[EFW07] A. Eskin, D. Fisher, and K. Whyte, Quasi-isometries and rigidity of solvable groups,Pure Appl. Math. Q. 3 (2007), no. 4, Special Issue: In honor of Grigory Margulis.Part 1, 927–947.

[EFW12] , Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric toCayley graphs, Ann. of Math. (2) 176 (2012), no. 1, 221–260.

[EFW13] , Coarse differentiation of quasi-isometries II: Rigidity for Sol and lamplightergroups, Ann. of Math. (2) 177 (2013), no. 3, 869–910.

[Ele17] G. Elek, On uniformly recurrent subgroups of finitely generated groups,arXiv:1702.01631 (2017).

[Ers06] A. Erschler, Generalized wreath products, Int. Math. Res. Not. (2006), Art. ID 57835,14.

[Eym72] P. Eymard, Moyennes invariantes et représentations unitaires, Lecture Notes in Math-ematics, Vol. 300, Springer-Verlag, Berlin-New York, 1972.

[FST15] J. Frisch, T. Schlank, and O. Tamuz, Normal amenable subgroups of the automorphismgroup of the full shift, arXiv:1512.00587 (2015).

[Fur67a] H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Dio-phantine approximation, Math. Systems Theory 1 (1967), 1–49.

[Fur67b] , Poisson boundaries and envelopes of discrete groups, Bull. Amer. Math. Soc.73 (1967), 350–356.

[Fur73] , Boundary theory and stochastic processes on homogeneous spaces, Harmonicanalysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, WilliamsColl., Williamstown, Mass., 1972), Amer. Math. Soc., Providence, R.I., 1973, pp. 193–229.

[Fur81] , Rigidity and cocycles for ergodic actions of semisimple Lie groups (after G. A.Margulis and R. Zimmer), Bourbaki Seminar, Vol. 1979/80, Lecture Notes in Math.,vol. 842, Springer, Berlin-New York, 1981, pp. 273–292.

[Fur01] A. Furman, Mostow-Margulis rigidity with locally compact targets, Geom. Funct. Anal.11 (2001), no. 1, 30–59.

[Fur03] , On minimal strongly proximal actions of locally compact groups, Israel J.Math. 136 (2003), 173–187.

[Ghy01] É. Ghys, Groups acting on the circle, Enseign. Math. (2) 47 (2001), no. 3-4, 329–407.[Gla74] S. Glasner, Topological dynamics and group theory, Trans. Amer. Math. Soc. 187

(1974), 327–334.[Gla75] , Compressibility properties in topological dynamics, Amer. J. Math. 97 (1975),

148–171.[Gla76] , Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag,

Berlin-New York, 1976.[Gru57] K. W. Gruenberg, Residual properties of infinite soluble groups, Proc. London Math.

Soc. (3) 7 (1957), 29–62.[Gui73] Y. Guivarc’h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc.

Math. France 101 (1973), 333–379.[GW15] E. Glasner and B. Weiss, Uniformly recurrent subgroups, Recent trends in ergodic the-

ory and dynamical systems, Contemp. Math., vol. 631, Amer. Math. Soc., Providence,RI, 2015, pp. 63–75.

[Har07] P. de la Harpe, On simplicity of reduced C∗-algebras of groups, Bull. Lond. Math. Soc.39 (2007), no. 1, 1–26.

[HR79] E. Hewitt and K. Ross, Abstract harmonic analysis. Vol. I, second ed., Grundlehren derMathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],


vol. 115, Springer-Verlag, Berlin-New York, 1979, Structure of topological groups,integration theory, group representations.

[Jen73] J. W. Jenkins, Growth of connected locally compact groups, J. Functional Analysis 12(1973), 113–127.

[JR00] P. Jolissaint and G. Robertson, Simple purely infinite C∗-algebras and n-filling actions,J. Funct. Anal. 175 (2000), no. 1, 197–213.

[Kaw17] T. Kawabe, Uniformly recurrent subgroups and the ideal structure of reduced crossedproducts, arXiv:1701.03413 (2017).

[Ken15] M. Kennedy, Characterizations of C∗-simplicity, arXiv:1509.01870v3 (2015).[KK17] M. Kalantar and M. Kennedy, Boundaries of reduced C∗-algebras of discrete groups,

J. Reine Angew. Math. 727 (2017), 247–267.[LB16] A. Le Boudec, Groups acting on trees with almost prescribed local action, Comment.

Math. Helv. 91 (2016), no. 2, 253–293.[LB17] , C∗-simplicity and the amenable radical, Invent. Math. 209 (2017), no. 1,

159–174.[LBMB16] A. Le Boudec and N. Matte Bon, Subgroup dynamics and C∗-simplicity of groups of

homeomorphisms, Ann. Sci. Ecole Norm. Sup. (to appear) (2016).[LBW16] A. Le Boudec and Ph. Wesolek, Commensurated subgroups in tree almost automor-

phism groups, Groups Geom. Dyn. (to appear) (2016).[Los87] V. Losert, On the structure of groups with polynomial growth, Math. Z. 195 (1987),

no. 1, 109–117.[LS96] M. Laca and J. Spielberg, Purely infinite C∗-algebras from boundary actions of discrete

groups, J. Reine Angew. Math. 480 (1996), 125–139.[Mal40] A. Malcev, On isomorphic matrix representations of infinite groups, Rec. Math. [Mat.

Sbornik] N.S. 8 (50) (1940), 405–422.[Mar91] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathe-

matik und ihrer Grenzgebiete (3), vol. 17, Springer-Verlag, Berlin, 1991.[Mar00] G. Margulis, Free subgroups of the homeomorphism group of the circle, C. R. Acad.

Sci. Paris Sér. I Math. 331 (2000), no. 9, 669–674.[MB18] N. Matte Bon, Rigidity of graphs of germs and homomorphisms between full groups,

arXiv:1801.10133 (2018).[MBT17] N. Matte Bon and T. Tsankov, Realizing uniformly recurrent subgroups,

arXiv:1702.07101 (2017).[MP03] N. Monod and S. Popa, On co-amenability for groups and von Neumann algebras, C.

R. Math. Acad. Sci. Soc. R. Can. 25 (2003), no. 3, 82–87.[MS04] N. Monod and Y. Shalom, Cocycle superrigidity and bounded cohomology for negatively

curved spaces, J. Differential Geom. 67 (2004), no. 3, 395–455.[MZ55] D. Montgomery and L. Zippin, Topological transformation groups, Interscience Pub-

lishers, New York-London, 1955.[Nek13] V. Nekrashevych, Finitely presented groups associated with expanding maps,

arXiv:1312.5654v1 (2013).[PV91] I. Pays and A. Valette, Sous-groupes libres dans les groupes d’automorphismes d’arbres,

Enseign. Math. (2) 37 (1991), no. 1-2, 151–174.[Rad17] N. Radu, New simple lattices in products of trees and their projections,

arXiv:1712.01091 (2017), With an appendix by P-E. Caprace.[Rag72] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York-

Heidelberg, 1972, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68.[Rat04] D. Rattaggi, Computations in groups acting on a product of trees: Normal sub-

group structures and quaternion lattices, ProQuest LLC, Ann Arbor, MI, 2004, Thesis(Dr.sc.math.)–Eidgenoessische Technische Hochschule Zuerich (Switzerland).

[Rém99] B. Rémy, Construction de réseaux en théorie de Kac-Moody, C. R. Acad. Sci. ParisSér. I Math. 329 (1999), no. 6, 475–478.

[Sha00] Y. Shalom, Rigidity of commensurators and irreducible lattices, Invent. Math. 141(2000), no. 1, 1–54.

[Tit70] J. Tits, Sur le groupe des automorphismes d’un arbre, Essays on topology and relatedtopics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 188–211.

44 ADRIEN LE BOUDEC

[Vor12] Y. Vorobets, Notes on the Schreier graphs of the Grigorchuk group, Dynamical systemsand group actions, Contemp. Math., vol. 567, Amer. Math. Soc., Providence, RI, 2012,pp. 221–248.

[Wis96] D. Wise, Non-positively curved squared complexes: Aperiodic tilings and non-residuallyfinite groups, ProQuest LLC, Ann Arbor, MI, 1996, Thesis (Ph.D.)–Princeton Univer-sity.

UCLouvain, IRMP, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium

CNRS, Unité de Mathématiques Pures et Appliquées, ENS-Lyon, FranceE-mail address: [email protected]

perso.uclouvain.be · AMENABLE UNIFORMLY RECURRENT SUBGROUPS AND LATTICE EMBEDDINGS ADRIEN LE...

Documents

Transcript of perso.uclouvain.be · AMENABLE UNIFORMLY RECURRENT SUBGROUPS AND LATTICE EMBEDDINGS ADRIEN LE...