Commensurated subgroups and micro-supported actions
Pierre-Emmanuel Caprace, Adrien Le Boudec, Dominik Francoeur
aa r X i v : . [ m a t h . G R ] F e b Commensurated subgroupsand micro-supported actions
Pierre-Emmanuel Caprace ∗ and Adrien Le Boudec † with an appendix by Dominik Francoeur ‡ UCLouvain, 1348 Louvain-la-Neuve, BelgiumUMPA - ENS Lyon, FranceFebruary 7, 2020
Abstract
Let Γ be a finitely generated group and X be a minimal compact Γ -space. Weassume that the Γ -action is micro-supported, i.e. for every non-empty open subset U ⊆ X , there is an element of Γ acting non-trivially on U and trivially on thecomplement X \ U . We show that, under suitable assumptions, the existence ofcertain commensurated subgroups in Γ yields strong restrictions on the dynamicsof the Γ -action: the space X has compressible open subsets, and it is an almost Γ -boundary. Those properties yield in turn restrictions on the structure of Γ : Γ isneither amenable nor residually finite. Among the applications, we show that the(alternating subgroup of the) topological full group associated to a minimal and ex-pansive Cantor action of a finitely generated amenable group has no commensuratedsubgroups other than the trivial ones. Similarly, every commensurated subgroup ofa finitely generated branch group is commensurate to a normal subgroup; the lat-ter assertion relies on an appendix by Dominik Francoeur, and generalizes a resultof Phillip Wesolek on finitely generated just-infinite branch groups. Other appli-cations concern discrete groups acting on the circle, and the centralizer lattice ofnon-discrete totally disconnected locally compact (tdlc) groups. Our results rely, inan essential way, on recent results on the structure of tdlc groups, on the dynamics oftheir micro-supported actions, and on the notion of uniformly recurrent subgroups. ∗ F.R.S.-FNRS Senior Research Associate. [email protected] † CNRS Researcher. [email protected] ‡ [email protected] Introduction
Let Γ be a group. Twosubgroups Λ , Λ ′ ≤ Γ are commensurate if Λ ∩ Λ ′ has finite index in Λ and Λ ′ . A subgroup Λ ≤ Γ is commensurated in Γ if all Γ -conjugates of Λ arecommensurate, or equivalently if the Λ -action on the coset space Γ / Λ has fi-nite orbits. Obviously, every normal subgroup of Γ is commensurated, and sois every subgroup of Γ that is commensurate to a normal subgroup (e.g. thefinite subgroups, and the subgroups of finite index). In general commensuratedsubgroups need not be commensurate to a normal subgroup. For instance ev-ery normal subgroup of the group SL( n, Z [1 /p ]) is finite or finite index, but SL( n, Z ) is an infinite and infinite index commensurated subgroup. Also in gen-eral the problems of understanding the normal subgroups and of understandingthe commensurated subgroups of a group Γ are rather independent, and a com-plete understanding of the normal subgroups does not necessarily provide adescription of the commensurated subgroups. This is for instance illustrated bythe case of lattices in higher rank semisimple Lie groups: while Margulis normalsubgroup theorem says that every normal subgroup of Γ is finite or finite in-dex, the Margulis-Zimmer conjecture on the description of the commensuratedsubgroups of Γ remains open in general (see [38] for partial answers). See also§6.1.4 below for another illustration.The study of the commensurated subgroups of a group Γ is well-known tobe closely related to the study of dense homomorphisms Γ → G from Γ toa totally disconnected locally compact group ( tdlc group hereafter). Here wecall a homomorphism dense if it has dense image in G . Indeed, if Γ → G is sucha homomorphism, then the preimage in Γ of every compact open subgroup of G is a commensurated subgroup of Γ . Conversely if Λ ≤ Γ is a commensuratedsubgroup, then the Schlichting completion process provides a tdlc group Γ // Λ and a dense homomorphism Γ → Γ // Λ [37]. We refer to [38, Section 3] fora detailed introduction to Schlichting completions (see also [35] for additionalproperties).Let Γ be a group acting by homeomorphisms on a topological space X . Therigid stabilizer Rist Γ ( U ) of a subset U ⊆ X is the pointwise fixator in Γ of thecomplement of U in X . The action of Γ on X is micro-supported if Rist Γ ( U ) acts non-trivially on X for every non-empty open subset U of X .In this article we relate the existence of commensurated subgroups of a group Γ with the topological dynamics of the micro-supported actions of Γ on compactspaces. The following is a simplified version of our main result (see Theorem5.4 for a more comprehensive statement).2 heorem 1.1. Let Γ be a finitely generated group such that every proper quo-tient of Γ is virtually nilpotent, and let X be a compact Γ -space such that theaction of Γ on X is faithful, minimal and micro-supported. If Γ has a commen-surated subgroup which is of infinite index and which is not virtually containedin a normal subgroup of infinite index of Γ , then the following hold:(i) X has a non-empty open subset which is compressible by Γ . In particular Γ is monolithic, hence not residually finite.(ii) X is an almost Γ -boundary. In particular Γ is not amenable. We recall some terminology. A subgroup Λ ≤ Γ is virtually contained in Λ ′ ≤ Γ if Λ has a finite index subgroup that is contained in Λ ′ . A group Γ is monolithic if the intersection M of all non-trivial normal subgroups of Γ isnon-trivial. When this holds M is called the monolith of Γ .Let X be a compact Γ -space. A non-empty open subset U of X is com-pressible by Γ if there exists x ∈ X such that for every neighbourhood V of x , there exists γ ∈ Γ such that γ ( U ) ⊂ V . The action of Γ on X is stronglyproximal if the orbit closure of every probability measure on X contains aDirac measure, and the action of Γ on X is a (topological) boundary if it isminimal and strongly proximal [14]. We will also say that the action of Γ on X is almost strongly proximal if X admits a Γ -invariant clopen partition X = X ∪ · · · ∪ X d such that for each i , the action of Stab Γ ( X i ) on X i is stronglyproximal; and X is an almost boundary if X is minimal and almost stronglyproximal.Theorem 1.1, as well as some intermediate results that we establish towardsits proof, have several types of applications. The rest of this introduction isaimed at describing them. If Λ is a group acting on a compact space X ,the topological full group F (Λ , X ) is the group of homeomorphisms g of X suchthat for every x ∈ X there exist a neighbourhood U of x and an element γ ∈ Λ such that g ( y ) = γ ( y ) for every y ∈ U . Topological full groups were first studiedin detail by Giordano-Putnam-Skau [16] and Matui [30], and more recently in[21, 23, 24, 29] (see also the survey [13] for additional references and historicalremarks).We denote by A (Λ , X ) ≤ F (Λ , X ) the alternating full group introducedby Nekrashevych in [33]. If Λ is a finitely generated group and Λ y X isa minimal and expansive action on a Cantor space X , Nekrashevych showedthat the group A (Λ , X ) is the monolith of F (Λ , X ) , and is a finitely generatedand simple group. When Λ = Z the alternating full group coincides with the3erived subgroup of F (Λ , X ) , and in that case finite generation and simplicityof F (Λ , X ) ′ were previously obtained by Matui in [30]. We refer to Section 6.1for definitions and details.Applying Theorem 1.1, we obtain the following result: Theorem 1.2.
Let Λ be a finitely generated group, and Λ y X a minimal andexpansive action on a Cantor space X such that A (Λ , X ) y X does not admitany compressible open subset. Then every proper commensurated subgroup ofthe alternating full group A (Λ , X ) is finite. We recall that there exist examples of alternating full groups A (Λ , X ) suchthat the action of A (Λ , X ) on X admits compressible open subsets, and suchthat A (Λ , X ) admits commensurated subgroups that are infinite and of infiniteindex. This is for instance the case of the Higman-Thompson groups V d,k from[22], which admit a commensurated subgroup such that the associated Schlicht-ing completion is a Neretin group of almost automorphisms of a tree [7] (and itwas proven in [28] that V d,k admits infinitely commensurated subgroups). The-orem 1.2 does not say anything about such situations, but we prove in Theorem6.6 that for every Schlichting completion G of A (Λ , X ) , the action of A (Λ , X ) on X extends to a continuous action of G on X .An important situation where Theorem 1.2 applies is when X admits aprobability measure that is invariant under the action of Λ . Such a probabilitymeasure will also be invariant under the action of the full group F (Λ , X ) . Sincethe existence of an invariant probability measure prevents the existence of acompressible open subset, we have the following: Corollary 1.3.
Let Λ be a finitely generated group, and Λ y X be a minimaland expansive action on a Cantor space X . If X carries a Λ -invariant probabil-ity measure (e.g. if Λ is amenable), then every proper commensurated subgroupof the alternating full group A (Λ , X ) is finite. Corollary 1.3 implies that for every minimal and expansive action of Z d ona Cantor space, every proper commensurated subgroup of A ( Z d , X ) is finite.Recall that when d = 1 , Juschenko and Monod showed that the group F ( Z , X ) is amenable [23]. One motivation for studying specifically the commensuratedsubgroups of finitely generated infinite simple amenable groups comes from thefact that, if such a group Γ were known to admit an infinite proper commensu-rated subgroup Λ , then by results of [9], the Schlichting completion Γ // Λ wouldadmit as a quotient a compactly generated tdlc group that is non-discrete,topologically simple and amenable (see Proposition 3.6 in [12]). As of now, noexample of such group is available (see Question 3 in [12]). Hence Corollary 1.3implies that the above strategy to build such a group cannot work by starting4ith groups such as A (Λ , X ) . Corollary 1.3 also answers a question raised atthe end of [6]. Remark . The assumptions that Λ is finitely generated and that the actionof Λ on X is expansive are both essential; see Example 6.5. Remark . The conclusions of Theorem 1.2 and Corollary 1.3 hold more gen-erally for an arbitrary subgroup of F (Λ , X ) that contains A (Λ , X ) , with appro-priate modifications in the conclusion. See Theorem 6.4. As mentionned above, every subgroup of a group Γ thatis commensurate to a normal subgroup is commensurated. A group Γ in whichevery commensurated subgroup is commensurate to a normal subgroup maythus be viewed as a group with as few commensurated subgroups as possible.That property was called “inner commensurator-normalizer property” in [38].It is equivalent to the fact that every Schlichting completion of Γ is compact-by-discrete (see Lemma 5.1). Recall that a group G is compact-by-discrete if G admits a compact normal subgroup K such that G/K is discrete.We refer the reader to Section 6.2 for basic definitions and to [20, 3] for ageneral introduction to branch groups.
Theorem 1.6.
Let Γ be a finitely generated branch group. Then every com-mensurated subgroup of Γ is commensurate to a normal subgroup of Γ . Theorem 1.6 recovers and extends a result of Wesolek [41], who showed thesame result under the additionnal assumption that Γ is just-infinite (i.e. everynon-trivial normal subgroup of Γ is of finite index). We refer the reader to [15] for an intro-duction to group actions on the circle. The following result does not formallyfollow from our main result. Its proof rather relies on intermediate results thatwe obtain in the course of the proof of Theorem 5.4, suitably combined withthe Ghys–Margulis theorem about the topological dynamics of group actions onthe circle (see Theorem 6.9).
Theorem 1.7.
Let Γ be a finitely generated group with a faithful, minimal,micro-supported action on the circle, such that the subgroup of Γ generated bythe elements that fix pointwise an open interval has finite index in Γ . Thenevery commensurated subgroup of Γ is commensurate to a normal subgroup of Γ . See Example 6.12 for a family of groups to which this result applies. Inparticular, we recover the fact, first established in [27], that every proper com-mensurated subgroup of Thompson’s group T is finite.5 .5 Micro-supported actions of locally compact groups. Our resultsalso have applications to the study of non-discrete locally compact groups. Wedenote by S td the class of compactly generated tdlc groups that are non-discreteand topologically simple. We refer to [12, Appendix A] for a description ofvarious families of examples of groups within this class. Recall from [11] thatevery group G in S td (and more generally every tdlc group satisfying a conditioncalled [A]-semisimplicity, see Section 3.4) has an associated G -Boolean algebra LC ( G ) called the centralizer lattice of G , with the property that the action of G on the Stone space Ω G of LC ( G ) is minimal and micro-supported, and suchthat every minimal and micro-supported totally disconnected compact G -spaceis a factor of Ω G [11, Theorem 5.18]. In Section 6.4 we obtain a strengtheningof this universal property, by showing that Ω G is the maximal highly proximalextension in the sense of Auslander and Glasner [1] of any minimal and micro-supported totally disconnected compact G -space (Theorem 6.15).In Section 6.4 we also consider the situation of a pair of groups ( H, G ) suchthat there exists a continuous homomorphism ϕ : H → G with dense image.Examples of such pairs ( H, G ) naturally arise in the study of the local structureof groups in the class S td , see [34, Theorem 1.2] and [10]. In this situation, westudy the relationship between the centralizer lattice of H and that of G . Thefollowing is a simplified version of Theorem 6.17. Theorem 1.8.
Let
G, H ∈ S td and ϕ : H → G be a continuous injective homo-morphism with dense image. If Ω H is non-trivial, then the H -action on Ω G ismicro-supported, and there exists a H -map Ω H → Ω G that is a highly proximalextension. In particular Ω G is non-trivial. This result applies for instance when H is the commensurator in G of aninfinite pro- p Sylow subgroup of a compact open subgroup of G (Corollary 6.18).We refer to Section 6.4 for details. Acknowledgements.
We are grateful to Yves de Cornulier for his commentson an earlier version of this paper. This work was micro-supported by a PEPSgrant from CNRS and by the project ANR-19-CE40-0008-01 AODynG. We alsothank the organizers of the following conferences, and the support of the associ-ated host institutions:
Measurable, Borel, and Topological Dynamics in October2019 at CIRM, and
Groups, Dynamics, and Approximation in December 2019at the Mathematisches Forschungsinstitut Oberwolfach.6
Preliminaries
Let G be a group acting by homeomorphisms on a Haus-dorff space X . Recall that the action of G on X is micro-supported if the rigidstabilizer Rist G ( U ) acts non-trivially on X for every non-empty open subset U ⊆ X (by extension when G is clear from the context we will also say that the G -space X is micro-supported). Note that for faithful actions this is equivalentto saying that Rist G ( U ) is non-trivial. Note also that if the action of G on X ismicro-supported, then X cannot have isolated points.The following lemma is immediate from the definitions. Lemma 2.1.
Let G be a group, X, Y topological spaces on which G acts, and π : Y → X a continuous surjective G -equivariant map. Suppose that the actionon Y is micro-supported and the action on X is faithful. Then the action on X is micro-supported.Proof. If U is a non-empty open subset of X , then V = π − ( U ) is also open andnon-empty, so it follows that Rist G ( V ) is non-trivial since Y is micro-supported.Since Rist G ( V ) ≤ Rist G ( U ) , it follows that Rist G ( U ) is non-trivial, and hence Rist G ( U ) acts non-trivially on X since the action on X is faithful.If G is a topological group and X, Y are compact G -spaces, a continuous G -equivariant map π : Y → X will be called a G -map . When π is onto, we saythat Y is an extension of X , and that X is a factor of Y . We will also saythat π : Y → X is an extension or a factor map. A G -map π : Y → X that isalso a homeomorphism will be called an isomorphism.A non-empty subset C ⊂ X is compressible if there exists x ∈ X suchthat for every neighbourhood U of x , there exists g ∈ G such that g ( C ) ⊂ U .Note that by definition a compressible subset is necessarily non-empty. In the sequel we willmake use of the notion of highly proximal extension [1]. Let G be a topologicalgroup, and X, Y compact G -spaces. An extension π : Y → X is highly prox-imal if for every non-empty open subset U ⊆ Y , there exists x ∈ X such that π − ( x ) ⊆ U . When X, Y are minimal, this is equivalent to saying that for some(every) x ∈ X , π − ( x ) is compressible in Y [1].Auslander and Glasner showed that every minimal compact G -space X ad-mits a highly proximal extension π : X ∗ → X with the property that for anyhighly proximal extension p : Y → X there exists a G -map p ′ : X ∗ → Y suchthat π = p ◦ p ′ , and such that any highly proximal extension of X ∗ is an iso-morphism. Such a X ∗ is necessarily unique up to isomorphism, and is calledthe maximal highly proximal extension of X [1].7ollowing [1], we say that X, Y are highly proximally equivalent , denoted X ∼ hp Y , if X and Y admit a common highly proximal extension. Since anyhighly proximal extension Z → X gives rise to an isomorphism between Z ∗ and X ∗ , being highly proximally equivalent is indeed an equivalence relation. Werefer to [1] for details. Lemma 2.2.
Let
X, Y be minimal compact G -spaces that are highly proximallyequivalent. Then X is an almost G -boundary if and only if Y is an almost G -boundary.Proof. By considering a common highly proximal extension, we see that it isenough to prove the statement for every highly proximal extension π : Y → X .Suppose that the G -action on X is almost strongly proximal, and let X = X ∪ · · · ∪ X d be a G -invariant clopen partition such that for each i , the actionof G i := Stab G ( X i ) on X i is a boundary action. Let Y i = π − ( X i ) . Then Y = Y ∪ · · · ∪ Y d is a G -invariant clopen partition of Y , and the restriction π | Y i : Y i → X i is a highly proximal extension between the G i -spaces Y i and X i .Therefore minimality of X i under the action of G i is inherited by Y i , and so isstrong proximality by [18, Lemma 5.2] (the proof is given there for almost one-to-one extensions, but the same argument applies to highly proximal extensions).Conversely, since minimality and strong proximality are inherited by factors,it is clear that if Y admits a clopen partition as above, then the push-forwardof this partition in X via π provides a partition of X (although distinct blocksmight have the same image) with the required properties.The following shows that the properties of being micro-supported and ofhaving compressible open subsets behave nicely with respect to highly proximalextensions. Proposition 2.3.
Let π : Y → X be a highly proximal extension betweencompact G -spaces. Then the following hold:(i) If A ⊆ X , then Rist G ( A ) ≤ Rist G ( π − ( A )) .(ii) X is micro-supported if and only if Y is micro-supported.If in addition X, Y are minimal, then:(iii) If A ⊆ X is compressible, then π − ( A ) is compressible.(iv) X admits a compressible open subset if and only if Y does. roof. (i) Let g ∈ Rist G ( A ) . Suppose for a contradiction that there exists y ∈ Y such that y / ∈ π − ( A ) and g ( y ) = y . Since π − ( A ) is closed, we may find an openneighbourhood W of y such that W ∩ g ( W ) = ∅ and W ∩ π − ( A ) = ∅ . Sincethe extension is highly proximal, there exists x ∈ X such that π − ( x ) ⊆ W , and x / ∈ A because x belongs to π ( W ) , which is disjoint from V . Moreover we have g ( x ) = x because g ( π − ( x )) ∩ π − ( x ) is empty, so we deduce that g / ∈ Rist G ( A ) ,which is our contradiction.(ii) Suppose X is micro-supported. Let U be a non-empty open subsetof Y . Denote by V the set of points x ∈ X such that π − ( x ) ⊆ U . Thesubset V is open, and V is non-empty since the extension is highly proximal.Let now W be a non-empty open subset of V such that W ⊂ V . We have π − ( W ) ⊆ π − ( V ) ⊆ U . According to (i), one has Rist G ( W ) ≤ Rist G ( U ) . Nowsince X is micro-supported, one can find x ∈ W and g ∈ Rist G ( W ) such that g ( x ) = x , and it follows that g ( y ) = y for every y ∈ π − ( x ) . So we have shownthat Rist G ( U ) acts non-trivially on U , and since U was arbitrary, it follows that Y is micro-supported.For the converse implication, we let V be a non-empty open subset of X ,and write U = π − ( V ) . If Y is micro-supported then one can find g ∈ Rist G ( U ) and W an open subset of U such that W and g ( W ) are disjoint. Then any point x ∈ V such that π − ( x ) ⊆ W is moved by g , and since g ∈ Rist G ( V ) , it followsthat Rist G ( V ) acts non-trivially on V , and X is micro-supported.(iii) Suppose that A ⊆ X is compressible, and write B = π − ( A ) . Let U be a non-empty open subset of Y , and let V the set of points x ∈ X such that π − ( x ) ⊆ U . Again V is open and non-empty, so using that A is compressiblein X and minimality, we find g ∈ G such that g ( A ) ⊆ V . It follows that g ( B ) ⊆ π − ( V ) ⊆ U . Since U was arbitrary, B is compressible.(iv) If U is a compressible open subset of X , then by V = π − ( U ) is acompressible open subset of Y by (iii). Conversely if V is compressible andopen in Y , then π ( V ) is a compressible subset of X . Now the image of anopen subset by a factor map between minimal compact G -spaces always hasnon-empty interior. So there exists a non-empty open subset W of X in π ′ ( V ) ,and it follows that W is a compressible open subset of X . A given group Γ may very well admit minimal andmicro-supported actions on compact spaces X and Y that are not isomorphic.For example Thompson’s group T admits such actions respectively on the circleand on a Cantor set [5]. More generally if Γ admits a minimal and micro-supported action on a Cantor set X , then by Proposition 2.3 the action of Γ on the maximal highly proximal extension X ∗ remains micro-supported, and X and X ∗ are never isomorphic, because X and X ∗ are not even homeomorphic9see below). However we have the following theorem of Rubin [36]. If X isa topological space, we denote by R ( X ) the Boolean algebra of regular opensubsets of X . Theorem 2.4.
Suppose that a group Γ admits a faithful and micro-supportedaction on topological spaces X, Y . Then there exists a Γ -equivariant isomor-phism R ( X ) → R ( Y ) . For a compact space X , we denote by ˜ X = S ( R ( X )) the Stone space of R ( X ) . Since R ( X ) is a complete Boolean algebra, ˜ X is a compact extremallydisconnected space (the closure of every open subset is open). There is a naturalmap π X : ˜ X → X , which associates to every ultrafilter on R ( X ) its limit inthe space X . This map is continuous and surjective, and has the property thatevery non-empty open subset of ˜ X contains a fiber π − ( x ) for some x ∈ X .Observe that every homeomorphism h of X induces a homeomorphism of ˜ X ,that we still denote h , such that π X ◦ h = h ◦ π X .If Γ is a discrete group and X a minimal compact Γ -space, the above dis-cussion says that ˜ X is also a minimal compact Γ -space, and π X : ˜ X → X is ahighly proximal extension. Moreover since ˜ X is extremally disconnected, it isactually the maximal highly proximal extension of X ([19, Lemma 2.3]). (See[43] for a generalization to non-discrete groups). The following is a consequenceof Rubin theorem and Proposition 2.3. Theorem 2.5.
Let Γ be a discrete group, and X a compact Γ -space that isfaithful and micro-supported. Then:(i) The action of Γ on ˜ X is faithful and micro-supported.(ii) For every compact Γ -space Y that is faithful and micro-supported, there isa factor map ˜ X → Y that is highly proximal.(iii) If X is minimal then ˜ X is minimal. Hence all compact Γ -space that arefaithful and micro-supported are minimal and highly proximally equivalent.(iv) If X is minimal and admits a compressible open subset, then all compact Γ -spaces that are faithful and micro-supported admit a compressible opensubset.Proof. The Γ -action on ˜ X is faithful because the Γ -action on X is faithful. Since π X : ˜ X → X is highly proximal and X is micro-supported, the action of Γ on ˜ X is micro-supported by Proposition 2.3. This shows (i).Let Y be a compact Γ -space that is faithful and micro-supported, and let π Y : ˜ Y → Y . If ϕ : ˜ X → ˜ Y is an isomorphism that is provided by the conclusionof Rubin theorem, then π Y ◦ ϕ is factor map ˜ X → Y that is highly proximal.So statement (ii) holds, and (iii) and (iv) follow from Proposition 2.3.10 Structure theory of tdlc groups
For a locally compact group G , we denote by B ( G ) the char-acteristic subgroup of G consisting of the elements with a relatively compactconjugacy class.We will invoke the following result from [40]. Theorem 3.2. If G is compactly generated tdlc group and G = B ( G ) , then G has a compact open normal subgroup. We will also need the following, see [32].
Theorem 3.3. If G is compactly generated tdlc group, then B ( G ) is a closedsubgroup of G . Given a tdlc group G , we denote by QZ( G ) the set ofelements whose centralizer is open. This is a (possibly non-closed) topologicallycharacteristic subgroup of G called the quasi-center . It contains all discretenormal subgroups of G . Moreover, if H is an open subgroup of G , we have QZ( H ) ≤ QZ( G ) .Given a closed subgroup H of a tdlc group G , we denoe by C G ( H ) thecentralizer of H in G . We also denote by QC G ( H ) the subgroup of G consistingof those g ∈ G such that C H ( g ) is relatively open in H . In particular, if H isopen we have QC H ( H ) = QZ( H ) and QC G ( H ) = QZ( G ) . We saythat a locally compact group G is just-non-compact if G is not compact andevery proper quotient of G by a closed normal subgroup is compact. Just-non-compact groups arise naturally as quotient groups of compactly generatednon-compact tdlc groups. Proposition 3.4.
Let G be a non-compact, compactly generated tdlc group.Then G has a closed normal subgroup N such that G/N is just-non-compact.If in addition G does not have any non-trivial finite quotient, then the quotient G/N is compactly generated and topologically simple.Proof.
The first assertion is [9, Proposition 5.2]. The second assertion followsfrom the first, together with the description of the structure of compactly gen-erated just-non-compact tdlc groups from [9, Theorem E].11ote however that the just-non-compact quotient group afforded by Proposi-tion 3.4 can in general be discrete. In fact, every compactly generated tdlc groupwith an infinite discrete quotient admits a just-non-compact discrete quotient(because every finitely generated infinite group admits a just-infinite quotient).In order to deal with discrete quotients, we need additional terminology.Given a locally compact group G , the discrete residual of G , denoted by Res( G ) , is defined as the intersection of all open normal subgroups of G . Noticethat Res( G ) is a closed topologically characteristic subgroup of G . We say that G is residually discrete if Res( G ) = 1 . The following result is due to G. Willis[42]. Proposition 3.5.
Let G be a compactly generated tdlc group. If G is virtuallynilpotent, then G has a basis of identity neighbourhoods consisting of compactopen normal subgroups. In particular G is residually discrete and compact-by-discrete. The goal of this section is to establish Theorem 3.13, which ensures theexistence of a specific kind of quotient groups for compactly generated tdlcgroups whose discrete residual is also compactly generated. This requires severalintermediate results that are of independent interest.
Proposition 3.6 ([9, Cor. 4.1]) . A compactly generated tdlc group is residuallydiscrete if and only if its compact open normal subgroups form a basis of identityneighbourhoods.
Recall that a locally compact group G is locally elliptic if every compactsubset of G generates a compact subgroup of G . The locally elliptic radical (LE-radical) of G is the largest closed normal subgroup of G that is locallyelliptic. Proposition 3.7.
Let G be a tdlc group. If G and Res( G ) are both compactlygenerated, then the following assertions hold, where R = G and for all n ≥ ,the group R n +1 is defined as R n +1 = Res( R n ) .(i) R /R is compact.(ii) R n = R for all n ≥ .(iii) R n is compactly generated for all n .Proof. By hypothesis, the groups R and R are compactly generated. Moreoverthe group R /R is residually discrete, so that by Proposition 3.6, there exists anopen normal subgroup N of G containing R and such that N /R is compact.12ince R is compactly generated, the quotient R /R is compactly generatedand residually discrete. It then follows from Proposition 3.6 that R /R has acompact open normal subgroup. In particular, there exists a closed normalsubgroup N of G with R ≤ N ≤ R such that N /R is the LE-radicalof R /R . Since R /N is discrete, the quotient group N /N is compactlygenerated and discrete-by-compact. Therefore N /N is compact-by-discrete(see [2, Lemma 4.4]). It follows that G/N is compact-by-discrete. In particular Res(
G/N ) is compact. Since N ≤ Res( G ) = R , we have Res(
G/N ) =Res( G ) /N . Therefore R /N is compact. Since R is compactly generated,we infer that N is compactly generated as well. It follows that the locallyelliptic group N /R is compactly generated, hence compact. Therefore R /R is compact, and R is also compactly generated.It remains to show that R = R . We know that R /R is compactlygenerated and residually discrete. Repeating the arguments of the previousparagraph using , we construct a closed normal subgroup N of G with R ≤ N ≤ R and such that N /R is the LE-radical of R /R . Since R /N isdiscrete and since N /R and R /R are both compact, we infer that N /N is compactly generated and discrete-by-compact. Therefore N /N is compact-by-discrete. As above, this implies that R /N is compact, so that R /N isalso compact, and N is compactly generated. This implies in turn that thelocally elliptic group N /R is compactly generated, hence compact. This finallyimplies that R /R is compact. Therefore R /R is compact, hence profinite, sothat Res( R /R ) = 1 . Since R ≤ R = Res( R ) , we deduce that Res( R ) /R =1 . In other words, we have R = R , as required.A special situation where the hypotheses of Proposition 3.7 are satisfied isdescribed in the following. Corollary 3.8.
Let G be a tdlc group all of whose open normal subgroups arecompactly generated. Then G and Res( G ) are compactly generated; in particularthe conclusions of Proposition 3.7 hold.Proof. Clearly G is compactly generated by hypothesis. Since G/ Res( G ) isresidually discrete, Proposition 3.6 ensures that there exists an open normalsubgroup N of G containing Res( G ) and such that N/ Res( G ) is compact. Inparticular Res( G ) is compactly generated, since N has this property by hypoth-esis.We shall prove the following result, which is a slight strengthening of [9,Proposition II.1]. A normal subgroup N of G is maximal if N is proper andevery normal subgroup M of G containing N is equal to N or G ; and N is13 inimal if N is non-trivial and every normal subgroup M of G contained in N is equal to or N . Proposition 3.9.
Let G be a non-trivial compactly generated tdlc group with Res( G ) = G . Let Max (resp.
Min ) be the collection of maximal (minimal)closed normal subgroups of G . Assume that T Max = 1 . Then:(i)
Min and
Max are finite and non-empty.(ii) The assignment N C G ( N ) establishes a bijective correspondence from Min to Max . Moreover N is the unique element of Min that is notcontained in C G ( N ) .(iii) The product of all elements of Min is dense in G .(iv) Every element of Min is a non-discrete compactly generated topologicallysimple tdlc group.Proof.
The hypotheses imply that
Max is non-empty. Moreover, for every M ∈ Max , the quotient
G/M is topologically simple and non-discrete since G = Res( G ) .We claim that G has a trivial center. Indeed, if z were a non-trivial element in Z ( G ) , then there would exist M ∈ Max not containing z , because T Max = 1 .The image of z in G/M would be a non-trivial central element, contradictingthat
G/M is topologically simple and non-discrete (in particular infinite, thusnon-abelian).All the hypotheses of Proposition II.1 in [9] are thus fulfilled. We deducethat the assertions (i), (iii) and the first assertion of (ii) hold.Let N ∈ Min . The group N maps onto a dense normal subgroup of thetopologically simple quotient group G/M . As observed in the proof of Propo-sition II.1 in [9], the group R has trivial quasi-center and trivial LE-radical. Inparticular the group N is non-abelian, non-compact and non-discrete. It followsthat M ∩ N = 1 , where M = C G ( N ) . Since M ∈ Max by (ii), we infer that N embeds as a dense normal subgroup of G/M .Any N ′ ∈ Max not contained in M commutes with M . If there existedsuch a group N ′ distinct from N , it would commute with N . Thus N and N ′ would be two commuting subgroups of G , both of which embed as dense normalsubgroup of G/M . This is implossible, because a Hausdorff group containingtwo commuting dense subgroups must be abelian, whereas
G/M is topologicallysimple and non-discrete, hence non-abelian. This confirms that N is the uniqueelement of Min that is not contained in C G ( N ) .Let π : G → G/M be the canonical projection, let U be a compact open sub-group of G and let V = N ∩ U . Then ϕ ( U ) is a compact open subgroup of G/M ,14nd ϕ ( V ) is a closed normal subgroup of ϕ ( U ) . Since N is a minimal normalsubgroup of G , it is generated by the G -conjugacy class of V . Equivalently, thegroup ϕ ( N ) is generated by the ( G/M ) -conjugacy class of ϕ ( V ) . By Proposi-tion 4.1(ii) in [12], there is a finite subset B ⊂ ϕ ( N ) such that G/M = h B i ϕ ( U ) .Since ϕ ( U ) normalizes ϕ ( V ) , it follows that the group h B i acts transitively onthe ( G/M ) -conjugacy class of ϕ ( V ) . Therefore ϕ ( N ) is generated by B ∪ ϕ ( V ) .Equivalently N is generated by V ∪ ( ϕ − ( B ) ∩ N ) , which is compact. Thus N is compactly generated.Since N ∈ Min , it is topologically characteristically simple. We may there-fore invoke [9, Corollary D], ensuring that N has closed normal subgroups S , . . . , S k which are topologically simple, and such that N = S . . . S k . Foreach i , observe that ϕ ( S i ) is a non-trivial subgroup of G/M normalized by ϕ ( N ) , which is dense. Since G/M is topologically simple, we infer that ϕ ( S i ) is dense. Since S i ∩ S j = 1 for i = j , we see that S i and S j commute. Since aHausdorff group containing two commuting dense subgroups must be abelian,we deduce that k = 1 . Therefore N is topologically simple. Remark . Let G be a compactly generated, topologically characteristicallysimple tdlc group. If G is non-compact and non-discrete, then Proposition 3.6implies that Res( G ) = G . Moreover [9, Theorem A] ensures that the set Max of maximal proper closed normal subgroups of G is non-empty. Hence T Max is trivial since G is topologically characteristically simple. Therefore Proposi-tion 3.9 applies to G . This refines slightly the description in [9, Corollary D(iv)]by ensuring that the minimal normal subgroups of G are compactly generated.A locally compact group G is monolithic if the intersection M of all non-trivial closed normal subgroups of G is non-trivial. In that case M is called the monolith of G . Note that M is topologically characteristically simple, so thatby the previous remark we have the following. Proposition 3.11.
Let G be compactly generated, monolithic, tdlc group. If themonolith M of G is non-discrete, non-compact and compactly generated, thenthe group M satisfies the conclusions of Proposition 3.9. The following observation shows that monolithic groups arise naturally asquotients of groups admitting minimal normal subgroups.
Lemma 3.12.
Let G be a locally compact group and N be a minimal non-trivialnormal subgroup of G . If N is non-abelian, then G/C G ( N ) is monolithic, withmonolith C G ( N ) N /C G ( N ) .Proof. (Compare the proof of Corollary 3.3 in [12].) Since N is non-abelian, it isnot contained in C G ( N ) , so that C G ( N ) N /C G ( N ) is a non-trivial closed normal15ubgroup of G/C G ( N ) . In order to complete the proof, we must show that everyclosed normal subgroup M of G containing C G ( N ) as a proper subgroup, alsocontains N . If the inclusion C G ( N ) < M is strict, then M does not commutewith N and, hence M ∩ N = 1 . Since N is minimal, it follows that N ≤ M , asrequired. Theorem 3.13.
Let G be a compactly generated tdlc group such that Res( G ) is also compactly generated. If G is not compact-by-discrete, then G has aclosed normal subgroup N such that the quotient H = G/N enjoys the followingproperties.(i) H is monolithic, and its monolith M is compactly generated, non-compactand non-discrete.(ii) M has closed normal subgroups S , . . . , S k which are compactly generated,topologically simple and non-discrete, and such that M = S . . . S k .(iii) H/M is compact-by-discrete.Proof.
We start by invoking Proposition 3.7. This ensures that R = Res(Res( G )) is a compactly generated closed normal subgroup of G such that G/R is compact-by-discrete and
Res( R ) = R . Since G is not compact-by-discrete, we deducethat R is non-trivial.It follows from [9, Theorem A] that R admits a maximal proper closed normalsubgroup M . Let L = T g ∈ G gM g − . Set G ′ = G/L , M ′ = M/L and R ′ = R/L .Then R ′ = Res(Res( G ′ )) and Res( R ′ ) = R ′ . Moreover G ′ /R ′ ∼ = G/R is compact-by-discrete. Furthermore, by the definition of R ′ , we see that the intersectionof all maximal proper closed normal subgroups of R ′ is trivial. It follows thatthe structure of R ′ is subjected to Proposition 3.9. By the definition of G ′ , theintersection of all G ′ -conjugates of M ′ is trivial. In view of Proposition 3.9(ii),it follows that every maximal proper closed normal subgroup of R ′ is conjugateto M ′ in G ′ . Since every proper closed normal subgroup of R ′ is contained in amaximal one (by Zorn’s lemma, using that R ′ is compactly generated), it followsthat the only closed normal subgroup of G ′ that is properly contained in R ′ isthe trivial subgroup. In other words, we have established that R ′ is a minimalnon-trivial closed normal subgroup of G ′ .We finally set H = G ′ /C G ′ ( R ′ ) . Thus H is monolithic with monolith M := C G ′ ( R ′ ) R ′ /C G ′ ( R ′ ) by Lemma 3.12. Since H/M = G ′ /C G ′ ( R ′ ) (cid:30) C G ′ ( R ′ ) R ′ /C G ′ ( R ′ ) ∼ = G ′ /C G ′ ( R ′ ) R ′ , we infer that H/M is isomorphic to a quotient of G ′ /R ′ . Since G ′ /R ′ is compact-by-discrete, and since every Hausdorff quotient of a compact-by-discrete group16s itself compact-by-discrete, we deduce that H/M is compact-by-discrete. Since R ′ is a quotient of R , it is compactly generated. Since it maps onto a densesubgroup of M , it follows that M is compactly generated. Since Res( R ′ ) = R ′ ,we deduce that Res( M ) = M . In particular M possesses maximal proper closednormal subgroups by Theorem A in [9]. Since M is minimal normal in H , theintersection of all its maximal proper closed normal subgroups is trivial, andthe remaining assertion to be established follows from Proposition 3.9. Let G be a tdlc group. A subgroup of G whosenormalizer is open is called locally normal . Following [11, 12], we say that G is [A]-semisimple if QZ( G ) = { } and if the only abelian locally normalsubgroup of G is the trivial subgroup. The notation QC G ( M ) has been recalledin Section 3.2. Proposition 3.14.
Let G be a monolithic tdlc group, and assume that themonolith M is compactly generated, non-compact and non-discrete. Then G are M are [A]-semisimple, and QC G ( M ) = 1 .Proof. Since M is a minimal normal subgroup of G , it is characteristically sim-ple. Since M is non-compact and non-discrete, it follows from [12, Proposi-tion 5.6] that QZ( M ) = 1 . We may then invoke [11, Proposition 6.17], whichimplies, together with [12, Proposition 5.6], that M is [A]-semisimple.The rest of the proof follows the reasoning in the proof of [10, Proposi-tion 5.1.2]. By [10, Proposition 4.4.3], we have QC G ( M ) = C G ( M ) . Since M is the monolith of G , either C G ( M ) = 1 or M ≤ C G ( M ) . The secondcase is excluded, because QZ( M ) = 1 . Therefore QC G ( M ) = 1 . In particular QZ( G ) = 1 .Let now A be an abelian compact locally normal subgroup of G . Let U be a compact open subgroup of N G ( A ) . Upon replacing U by AU , we mayassume that A is contained in U . Since A ∩ M is an abelian compact locallynormal subgroup of M , and since M is [A]-semisimple, we have A ∩ M = 1 . Inparticular A ∩ M ∩ U = 1 . Thus A and M ∩ U are normal subgroups of U withtrivial intersection, hence they commute. It follows that A centralizes an opensubgroup of M . Hence A ≤ QC G ( M ) = 1 .Two closed subgroups K, L of a tdlc group G are called locally equivalent if K ∩ L is relatively open in both K and L . Following [11], we define the structure lattice LN ( G ) of a tdlc group G as the set of local classes of closedlocally normal subgroups of G . The class of a locally normal subgroup K will be denoted [ K ] . Note that the group G acts by conjugation on LN ( G ) .The structure lattice LN ( G ) contains a canonical subset LC ( G ) , called the17 entralizer lattice , consisting of the local classes of centralizers of locallynormal subgroups of G . It was shown in [11] that the map ⊥ : LC ( G ) → LC ( G ) , [ K ] ⊥ = [ C G ( K )] , is well-defined and the operations [ K ] ∧ [ L ] = [ K ∩ L ] and [ K ] ∨ [ L ] = (cid:0) [ K ] ⊥ ∧ [ L ] ⊥ (cid:1) ⊥ turn LC ( G ) into a Boolean algebra. In addition we have the following result(see Theorem 5.2 and Theorem 5.18 in [11] ). The Stone space of a Booleanalgebra A will be denoted by S ( A ) . Theorem 3.15.
If a tdlc group G is [A]-semisimple, then the centralizer lattice LC ( G ) is a Boolean algebra, and the following hold:(i) If G acts faithfully on LC ( G ) , then the G -action on Ω G = S ( LC ( G )) iscontinuous and micro-supported.(ii) Every totally disconnected compact G -space on which the G -action is faith-ful and micro-supported is a G -factor of Ω G . The following result addresses the case where the G -action on LC ( G ) is notnecessary faithful. We recall that every Boolean algebra A is naturally endowedwith a partial order ≤ , defined by α ≤ β if α ∧ β = α for all α, β ∈ A . Thepartial order ≤ coincides with the relation of inclusion of clopen subsets of theStone space S ( A ) . Given a subset H of a group G , we denote by C G ( H ) the double centralizer of H in G , defined by C G ( H ) = C G ( C G ( H )) . Proposition 3.16.
Let G be an [A]-semisimple tdlc group, D ≤ G be a densesubgroup and Y be a totally disconnected compact G -space. Assume that the D -action on Y is micro-supported and faithful.Then the assignment α [ C G (Rist D ( α ))] defines an injective order-preservingmap from the Boolean algebra A of clopen subsets of Y into the centralizer lattice LC ( G ) .Proof. Let α ∈ A . Since the G -action on Y is continuous, the stabilizer G α isopen in G . In particular D α = D ∩ G α is dense in G α . Since D α normalizes Rist D ( α ) , we infer that Rist D ( α ) is normalized by G α . In particular it is alocally normal subgroup of G .Therefore C G (Rist D ( α )) = C G (Rist D ( α )) is a closed locally normal subgroupas well, and so is also C G (Rist D ( α )) . By the definition of LC ( G ) (see [11,Definition 5.1]), the local classes [ C G (Rist D ( α ))] and [ C G (Rist D ( α ))] both belongto LC ( G ) . 18or α ∈ A , we set f ( α ) := [ C G (Rist D ( α ))] ∈ LC ( G ) . We claim that if α is non-empty, then f ( α ) is a non-zero element of LC ( G ) .If f ( α ) = 0 , then C G (Rist D ( α )) is discrete, and thus contained in the quasi-center QZ( G ) , which is trivial since G is [A]-semisimple by hypothesis. Since C G (Rist D ( α )) contains Rist D ( α ) , we deduce that Rist D ( α ) is trivial, hence α = ∅ since the D -action on Y is micro-supported. This proves the claim.Let now α, β ∈ A with α ≤ β . Then Rist D ( α ) ≤ Rist D ( β ) . Therefore wehave C G (Rist D ( α )) ≤ C G (Rist D ( β )) and hence f ( α ) ≤ f ( β ) . It follows that themap f : A → LC ( G ) is order-preserving. This implies in particular that for all β, γ ∈ A , we have f ( β ∩ γ ) ≤ f ( β ) ∧ f ( γ ) .We next claim that if β ∩ γ = ∅ , then f ( β ) ∧ f ( γ ) = 0 . Indeed, if β and γ aredisjoint, then Rist D ( β ) and Rist D ( γ ) commute. Thus Rist D ( β ) ≤ C G (Rist D ( γ )) ,hence C G (Rist D ( β )) ≤ C G ( C G (Rist D ( γ ))) . Thus the closed locally normalsubgroups C G (Rist D ( β )) and C G (Rist D ( γ )) commute. In view of [11, Theo-rem 3.19(iii)], we deduce that f ( β ) ∧ f ( γ ) = 0 .It remains to prove the injectivity of f . To this end, let α, β ∈ A with α = β .Upon swapping α and β , we may assume that α β . Equivalently there existsa non-empty γ ≤ α in A which is disjoint from β . Since β ∩ γ = ∅ , we have f ( β ) ∧ f ( γ ) = 0 by the previous claim. Since γ ≤ α we have f ( γ ) ≤ f ( α ) . Ifwe had f ( α ) = f ( β ) , we would have f ( γ ) = f ( γ ) ∧ f ( α ) = f ( γ ) ∧ f ( β ) = 0 .Since γ is non-empty, this would contradict the claim established above. Thus f ( α ) = f ( β ) , as required. The dynamics of the action of a compactly generated [A]-semisimplegroup G on the Stone space Ω G of the centralizer lattice of G has been studiedin [12, Theorem 6.19] under the assumption that the G -action on Ω G is faithful.Recall from Proposition 3.14 that a compactly generated monolithic tdlcgroup G whose monolith is compactly generated, non-compact and non-discreteis [A]-semisimple. In particular the centralizer lattice LC ( G ) is a Boolean al-gebra. Its Stone space, denoted by Ω G , is thus a totally disconnected compact G -space. Theorem 3.17.
Let G be a compactly generated monolithic tdlc group, andassume that the monolith M is compactly generated, non-compact and non-discrete. Let Ω G be the Stone space of LC ( G ) .(i) If LC ( G ) is infinite, then G -action on Ω G is faithful. oreover, any totally disconnected compact G -space X on which the G -action is faithful and micro-supported is a factor of Ω G , and enjoys the followingproperties:(ii) The M -action on X is micro-supported.(iii) The G -action on X has a compressible clopen subset.(iv) The G -action on X is minimal.(v) Let S , . . . , S d be the minimal non-trivial closed normal subgroups of M .For each i , let X i be the complement of the fixed-point-set of S i in X . Then X i is clopen and X = S di =1 X i is a clopen partition of X . Furthermore,the S i -action on X i is minimal, strongly proximal, micro-supported, andhas a compressible clopen subset. In particular M is not amenable.Proof. Recall that the fact the S i exist and form a finite set is guaranteedby Proposition 3.11. Moreover each S i is a non-discrete compactly generatedtopolgically simple group, and S · · · S d is a dense subgroup of M .We argue by contradiction and assume that the G -action on Ω G is not faith-ful. In particular M acts trivially on Ω G .For each i , we define ω i : LC ( G ) → { , } : [ K ] (cid:26) if S i ∩ C M ( K ) = 10 otherwise.Since M acts trivially on LC ( G ) , for every [ K ] ∈ LC ( G ) the group QC M ( K ) is equal to C M ( K ) and is a closed normal subgroup of M by [11, Theorem 3.19].In particular since S i is a minimal normal subgroup of M , either S i ≤ C M ( K ) or S i intersects C M ( K ) trivially. So ω i ([ K ]) = 0 if and only if S i commutes with K , if and only if S i ∩ K = 1 [11, Theorem 3.19].We claim that ω i is a homomorphism of Boolean algebras. By the previousparagraph we have ω i ([ K ] ⊥ ) = ω i ([ K ]) ⊥ . Hence it is enough to check that forall [ H ] , [ K ] ∈ LC ( G ) , ω i ([ H ] ∧ [ K ]) = ω i ([ H ∩ K ]) is equal to ω i ([ H ]) ∧ ω i ([ K ]) .Clearly if ω i ([ H ]) = 0 or ω i ([ K ]) = 0 , i.e. if S i commutes with H or S i commuteswith K , then S i commutes with H ∩ K and ω i ([ H ∩ K ]) = 0 . Moreover if ω i ([ H ]) = ω i ([ K ]) = 1 , then S i ∩ H ∩ K = 1 . Indeed otherwise S i ∩ H wouldcommute with K by [11, Theorem 3.19], and hence S i ∩ H would be abelianbecause by assumption S i commutes with C G ( K ) since ω i ([ K ]) = 1 . Hence S i ∩ H would be an abelian locally normal subgroup of S i , which is non-trivialbecause ω i ([ H ]) = 1 . This contradicts [12, Theorem 5.3]. So S i ∩ H ∩ K = 1 ,and ω i ([ H ∩ K ]) = 1 . This shows the claim.20ow for [ K ] ∈ LC ( G ) , K = 1 , the subgroup K cannot commute will all the S i , because otherwise K would commute in M and M has trivial centralizer. Sothere exists i such that ω i ([ K ]) = 1 . This means that every non-empty clopensubset of Ω G contains one ω i , so it follows that Ω G is trivial. Contradiction. Sothe action of G on Ω G is faithful.Let now X be a totally disconnected compact G -space on which the G -actionis faithful and micro-supported. By Theorem 3.15, the space X is a factor of Ω G . Equivalently, the centralizer lattice LC ( G ) contains a G -invariant Booleansubalgebra A whose Stone dual is G -equivariantly homeomorphic to X . Wemay thus identify the clopen subsets of X with the elements of A . Since the G -action on X is faithful, so is the action on A .For every non-empty clopen α of X , the rigid stabilizer Rist G ( α ) is a non-discrete closed locally normal subgroup of G . In view of Proposition 3.14,we may invoke [10, Proposition 7.1.2(i)] which ensures that Rist G ( α ) ∩ M =Rist M ( α ) is non-discrete. This proves (ii).By [10, Proposition 7.3.1], the G -action on X has a compressible clopensubset, thus (iii) holds.To prove (iv) and (v), we let X i be the complement of the fixed-point-set of S i in X , for each i ∈ { , . . . , d } . Since the G -action on X is faithful, so is the M -action, and we deduce that X i is non-empty. Moreover M is [A]-semisimpleby Proposition3.14. In view of (ii), we may therefore invoke [12, Theorem 6.19]for the M -action on X (the latter result requires that M is locally C-stable ,which is a consequence of [A]-semisimplicity by [11, Proposition 6.17]). Thisensures that the X i are pairwise disjoint.We next claim that for each i ∈ { , . . . , d } , the S i -action on X i is micro-supported and has a compressible clopen subset. Indeed, applying again [12,Theorem 6.19(i)] to the M -action on X , we obtain, for each i , a non-emptyclopen α i ⊂ X i such that for every non-empty clopen β in X , there exists i ∈ { , . . . , d } and g ∈ M with gα i ⊂ β . Let us now fix i ∈ { , . . . , d } and letus assume that the non-empty clopen β is contained in X i .Since the M -action is continuous and since the product S . . . S d is densein M , we can find g ∈ S . . . S d with gα i ⊂ β . Let us write g = g . . . g d with g i ∈ S i . Since β is contained in the fixed-point-set of g j for all j = i , and since X i ∩ X j = ∅ , we deduce that g i α i ⊂ β . This proves that α i is a compressibleclopen subset for the S i -action on the closure X i .Notice that Rist M ( α i ) is a non-trivial closed locally normal subgroup of M .Using [12, Theorem 6.19(iii)], we deduce that Rist S i ( α i ) is non-discrete. Since α i is a compressible clopen subset for the S i -action on X i , it follows that the S i -action on X i is micro-supported. This proves the claim.Since S i is a compactly generated, non-discrete, topologically simple tdlc21roup, we may invoke [12, Theorem J(ii)], which ensures that the S i -action on X i is minimal and strongly proximal. In particular S i does not fix any point of X i . Therefore X i = X i , thereby confirming that X i is clopen in X . Since M acts trivially on the complement of S i X i , we must have X = S di =1 X i becausethe G -action is faithful. Thus (v) holds.We finally observe that the clopen subsets X i are the minimal non-empty M -invariant closed subsets of X . In particular every non-empty G -invariantclosed subset contains X i for some i . Since M is the monolith of G , the G -action by conjugation permutes the set { S , . . . , S d } transitively. Therefore the G -action on X permutes the sets { X , . . . , X d } transitively. In view of (v), wededuce that (iv) holds. Remark . We emphasize that the compact generation of the monolith isessential for assertion (iv) in Theorem 3.17. An example illustrating this matterof fact is provided by the group G = Ner( T ) ξ , where T is the d -regular tree, d ≥ , ξ is an end of T , and Ner( T ) is the Neretin group of T (which acts on ∂T ). The G -action on ∂T is faithful and micro-supported. Moreover, the group G is monolithic, but its monolith is not compactly generated. The G -action on ∂T has a compressible clopen subset, but it is not minimal as it fixes ξ . The setting of the current section will be the fol-lowing:a) L is a locally compact group, and X is a compact L -space such that theaction of L on X is faithful, minimal and micro-supported.b) ϕ : L → G is a continuous injective homomorphism from L into a secondcountable tdlc group G , such that ϕ ( L ) is dense in G .We emphasize that, while the group G is typically non-discrete, the group L is allowed to be discrete. While the applications in Section 6 that will beobtained in §6.1-6.2-6.3 only deal with the case where L is a discrete group,the setting of §6.4 requires L to be non-discrete, whence this choice for thecurrent section. Notice that if the group L is generated by a subset S , then G is generated by ϕ ( S ) together with any neighbourhood of the identity because ϕ ( L ) is dense in G . In particular if L is compactly generated then G is compactlygenerated as well. 22 .2 Construction of a URS of G . Let G be a locally compact group, and de-note by Sub ( G ) the space of closed subgroups of G , endowed with the Chabautytopology. The space Sub ( G ) is compact, and is metrizable if G is second count-able. A closed minimal G -invariant subspace of Sub ( G ) is called a uniformlyrecurrent subgroup (URS) of G [17].Let X be a compact space. Recall that a map ψ : X → Sub ( G ) is lowersemi-continuous if for every open subset U ⊂ G , the set of x ∈ X such that ψ ( x ) ∩ U = ∅ is open in X . This is equivalent to saying that for every net ( x i ) in X converging to x and such that the net ( ψ ( x i )) converges to H , one has ψ ( x ) ⊆ H .In the sequel we consider L, X, ϕ, G as in the general setting. Observe thatthe group L has a continuous conjugation action on Sub ( G ) defined via ϕ . Lemma 4.1.
For
L, X, ϕ, G as in the general setting, the map ψ : X → Sub ( G ) , x ϕ ( L x ) , is lower semi-continuous and L -equivariant.Proof. Let x ∈ X and U an open subset of G such that ψ ( x ) ∩ U = ∅ . Then ϕ ( L x ) ∩ U = ∅ , and if γ ∈ L x is such that ϕ ( γ ) ∈ U , then by definition of L x there is an open neighbourhood V of x in X on which γ acts trivially. Then γ ∈ L y for every y ∈ V , and it follows that ϕ ( L y ) ∩ U = ∅ for every y ∈ V .This shows lower semi-continuity. That ψ is L -equivariant follows from thedefinitions.In the sequel we will denote by X ϕ ⊆ X the set of points where the map x ϕ ( L x ) is continuous. Since the group G is second countable, the space Sub ( G ) is metrizable, so by a general argument of semi-continuity it followsfrom Lemma 4.1 that X ϕ is a dense subset of X (it is comeager). In the sequelwe write F ϕ ( X ) := n(cid:16) x, ϕ ( L x ) (cid:17) : x ∈ X o ⊆ X × Sub ( G ) ,E ϕ ( X ) := n(cid:16) x, ϕ ( L x ) (cid:17) : x ∈ X ϕ o ⊆ F ϕ ( X ) ,T ϕ,G ( X ) := n ϕ ( L x ) : x ∈ X o and S ϕ,G ( X ) := n ϕ ( L x ) : x ∈ X ϕ o . Note that L acts diagonally on X × Sub ( G ) , and F ϕ ( X ) and E ϕ ( X ) areclosed and L -invariant. In the following statement we denote respectively by p , p the projections from X × Sub ( G ) to the first and second factor. Recallthat an extension π : Y → X between minimal compact G -spaces is almostone-to-one if the set of y ∈ Y such that π − ( π ( y )) = { y } is dense in Y . Notethat every almost one-to-one extension is highly proximal.23 roposition 4.2. The following hold:(i) E ϕ ( X ) is the unique minimal closed L -invariant subset of F ϕ ( X ) , and S ϕ,G ( X ) is the unique minimal closed G -invariant subset of T ϕ,G ( X ) . Inparticular S ϕ,G ( X ) is a URS of G .(ii) The extension p : E ϕ ( X ) → X is almost one-to-one, and one has p ( E ϕ ( X )) = S ϕ,G ( X ) .Proof. See Theorem 2.3 in [18]. The only additional observation that is neededhere is that S ϕ,G ( X ) is indeed G -invariant, but this is clear since S ϕ,G ( X ) is L -invariant and L has dense image in G . S ϕ,G ( X ) is infinite. The goal of this sectionis to exhibit certains conditions ensuring that the space S ϕ,G ( X ) constructedin Section 4.2 is not degenerate (i.e. is not a finite set), and also conditionsensuring that the action of G on S ϕ,G ( X ) is faithful.The following key lemma is the starting point of our discussion. Lemma 4.3.
Let Γ be a group with a faithful action on a Hausdorff space X .Let also ϕ : Γ → G be an injective homomorphism to a locally compact group G .Let x ∈ X and H be a subgroup of the centralizer C G ( ϕ (Γ x )) . If the closedsubgroup J = ϕ (Γ x ) H ≤ G is compactly generated, then there is a non-emptyopen subset V ⊂ X such that ϕ (Rist Γ ( V )) ≤ B ( J ) .Proof. Let us denote by F x the family of closed subsets of X not containing x .By definition one has Γ x = [ C ∈F x Rist L ( C ) . Since ϕ (Γ x ) H = J , we deduce that J belongs to the closure in Sub ( J ) of theset n ϕ (Rist Γ ( C )) H : C ∈ F x o . By hypothesis, the group J is compactly generated, and thus it admits aChabauty neighbourhood consisting of cocompact subgroups (see [4, VIII.5.3,Proposition 6]). Therefore we may find C ∈ F x such that ϕ (Rist Γ ( C )) H iscocompact in J . Let U be the complement of C in X . Since X is Hausdorff, wemay find a non-empty open subset V ⊂ U such that U \ V is a neighbourhood of x . Then Rist Γ ( V ) centralizes Rist Γ ( C ) because V and C are disjoint, and since H centralizes ϕ (Γ x ) and ϕ (Rist Γ ( V )) ≤ ϕ (Γ x ) , we deduce that ϕ (Rist Γ ( V )) centralizes ϕ (Rist Γ ( C )) H . Since the latter is cocompact in J , it folllows that ϕ (Rist Γ ( V )) ≤ B ( J ) . 24 roposition 4.4. Let
L, X, ϕ, G as in the general setting, and assume that G is compactly generated. Suppose also that the action of L on X has the propertythat for every closed subgroup H ≤ L such that the normalizer of H in L hasfinite index and such that there exists x ∈ X with L x ≤ H , then H must becocompact in L .If S ϕ,G ( X ) is finite, then there exists a non-empty open subset V ⊂ X suchthat ϕ (Rist L ( V )) ≤ B ( G ) .Proof. By assumption there exists a closed subgroup J ≤ G such that N G ( J ) has finite index in G and S ϕ,G ( X ) = (cid:8) J, g J g − , . . . , g n J g − n (cid:9) is the conjugacy class of J in G . Let H = ϕ − ( J ) . By definition of S ϕ,G ( X ) wemay find x ∈ X such that ϕ ( L x ) = J , so in particular L x ≤ H . Moreover thenormalizer of H in L has finite index in L , so it follows from our assumptionthat H must be cocompact in L . Since L has dense image in G , this impliesthat ϕ ( H ) is cocompact in G . It follows that J is also cocompact in G , so inparticular J is compactly generated. The conclusion follows from Lemma 4.3using the fact that B ( J ) ≤ B ( G ) because J is cocompact in G . Definition 4.5.
A group L is just non virtually nilpotent (j.n.v.n.) if L isnot virtually nilpotent and every proper quotient of L is virtually nilpotent. Definition 4.6.
The action of L on a minimal compact L -space X is stronglyjust-infinite if for every distinct points x, y in X , the subgroup of L generatedby L x and L y has finite index in L .Note that this condition implies that for every non-injective L -map X → Y that is a factor, then Y must be finite. This explains the choice of terminology. Proposition 4.7.
Let
L, X, ϕ, G as in the general setting, and assume that L is j.n.v.n., that the action of L on X is strongly just-infinite, and that the group G is compactly generated. If S ϕ,G ( X ) is finite, then G is compact-by-discrete.Proof. Since the action of L on X is strongly just-infinite, all the assumptionsof Proposition 4.4 are satisfied. So if S ϕ,G ( X ) is finite, then Proposition 4.4yields an element γ ∈ L such that ϕ ( γ ) ∈ B ( G ) . Let N = hh γ ii G . Thus N is acompactly generated closed normal subgroup of G contained in B ( G ) . It followsfrom Theorem 3.2 that N is compact-by-discrete. Since N ∩ ϕ ( L ) is non-trivial,it follows that the image of L in G/N is virtually nilpotent, and hence that
G/N is residually discrete by Proposition 3.5. Using [8, Proposition 2.2], we knowthat a compactly generated tdlc group that is discrete-by-{residually discrete}must be residually discrete, and hence compact-by-discrete by Proposition 3.6.This implies that G is compact-by-discrete, as desired.25 roposition 4.8. Let
L, X, ϕ, G as in the general setting. Assume that L is j.n.v.n. and that G is compactly generated, monolithic, with a non-discrete,non-compact, compactly generated monolith M , and such that G/M is compacy-by-discrete. Then the action of G on S ϕ,G ( X ) is faithful.Proof. By Proposition 3.11, the set of minimal normal subgroups of M is finiteand non-empty. Its elements are denoted by S , . . . , S d . Thus S i is a compactlygenerated, non-discrete, topologically simple tdlc group.Suppose the action of G on S ϕ,G ( X ) is not faithful. Then M acts trivially on S ϕ,G ( X ) . In the sequel we fix a point x such that ϕ ( L x ) belongs to S ϕ,G ( X ) . Thesubgroup M ∩ ϕ ( L x ) is normal in M because M normalizes ϕ ( L x ) . Therefore byminimlity of S , . . . , S d , for each i we have either S i ≤ ϕ ( L x ) or S i ∩ ϕ ( L x ) = 1 .Let I be the set of those i ∈ { , . . . , d } such that S i ∩ ϕ ( L x ) = 1 , and set H = h S i ∈ I S i i . Observe that H centralizes ϕ ( L x ) , and that M = S . . . S d isentirely contained in J = ϕ ( L x ) H .Recall that by assumption G/M is compact-by-discrete. Since G is not dis-crete and L has dense image in G , every discrete quotient of G is a properquotient of L , and hence is finitely generated virtually nilpotent. Therefore thegroup G/M is compactly generated and compact-by-{discrete finitely generatedvirtually nilpotent}, and it follows that every closed subgroup of
G/M is com-pactly generated. In particular
J/M is compactly generated, and thus also J since M is compactly generated. Lemma 4.3 therefore implies that B ( J ) is non-trivial. Since B ( M ) is trivial, we have B ( J ) ∩ M = 1 so that B ( J ) ≤ C G ( M ) .It follows that C G ( M ) is non-trivial. Since M is the monolith of G , we musthave M ≤ C G ( M ) , so M is abelian. In particular M is compact-by-discrete,and we have already seen above that this is prevented by the hypotheses. The proof of the following propo-sition is inspired by [26, §4.1.3] (see also [29, Proposition 7.18]).
Proposition 4.9.
Let
L, X, ϕ, G as in the general setting, and assume that theaction of L on X is strongly just-infinite, and that S ϕ,G ( X ) is infinite. Thenthe following hold:(i) There exists a L -map ψ : S ϕ,G ( X ) → X , and ψ is almost one-to-one.(ii) The action of L on X extends to an action of G , and the map ψ : S ϕ,G ( X ) → X is a G -map.(iii) If the G -action on S ϕ,G ( X ) is faithful then the G -action on X is faithful. roof. (i) For K ∈ S ϕ,G ( X ) , we define ψ ( K ) to be the unique x ∈ X suchthat ϕ ( L x ) ≤ K . In order to see that ψ is well-defined, we have to show thatsuch a point x exists and is unique. Recall that there exists a dense subset ofpoints K ∈ S ϕ,G ( X ) which are of the form ϕ ( L x ) for some x ∈ X . So for thesepoints the existence is clear, and the general case where K is arbitrary followsby semi-continuity (Lemma 4.1). Suppose now that there is K for which thispoint x is not unique. Then by the assumption that the action of L on X isstrongly just-infinite and the fact that L has dense image in G , it follows that K is actually a finite index subgroup of G . Hence K has a finite conjugacy classin G , which contradicts the assumption that S ϕ,G ( X ) is continuous. So x mustbe unique, and ψ is well-defined.By semi-continuity again, if ( K n ) converges to K in S ϕ,G ( X ) and x n = ψ ( K n ) converges to x , then ϕ ( L x ) ≤ K . But ψ ( K ) is the unique point with thisproperty, so it follows that x = ψ ( K ) . Hence the map ψ is continuous. Notethat ψ is clearly L -equivariant.To show that ψ is almost one-to-one, denote by X ϕ the dense G δ -set ofpoints where the map x ϕ ( L x ) is continuous (Proposition 4.2). Then weclaim that ψ − ( { x } ) = n ϕ ( L x ) o for all in X ϕ . Indeed, suppose that K is suchthat ϕ ( L x ) ≤ K . Since K is in S ϕ,G ( X ) , we may find a net ( x i ) in X suchthat ϕ ( L x i ) converges to K , and we may assume that ( x i ) converges to some y in X . By semi-continuity ϕ ( L y ) ≤ K , so it follows that actually y = x , andconsequently K = ϕ ( L x ) since x ∈ X ϕ . This proves the claim.(ii) The Gelfand correspondence establishes a bijection between L -invariant C ∗ -sub-algebras of C ( S ϕ,G ( X )) (the continuous complex valued functions on S ϕ,G ( X ) ) and L -equivariant factors of S ϕ,G ( X ) . The group G acts on S ϕ,G ( X ) ,and by density of L in G any L -invariant C ∗ -sub-algebra of of C ( S ϕ,G ( X )) mustbe G -invariant. Hence by duality G acts on X extending the action of L , andthe map ψ : S ϕ,G ( X ) → X is G -equivariant.Finally (iii) easily follows from the fact that ψ is almost one-to-one.The following result follows from the combination of Proposition 4.8 andProposition 4.9. Corollary 4.10.
Let
L, X, ϕ, G as in the general setting. Assume that L isj.n.v.n. and that the action of L on X is strongly just-infinite. Assume alsothat G is compactly generated, monolithic, with a non-discrete, non-compact,compactly generated monolith M , and such that G/M is compacy-by-discrete.Then the L -action on X extends to a faithful G -action. Commensurated subgroups and micro-supportedactions
Let Γ be a group and Λ ≤ Γ a commensu-rated subgroup. We denote by Γ // Λ the Schlichting completion of the pair (Γ , Λ) , defined as the closure of the natural image of Γ in the symmetric group Sym(Γ / Λ) endowed with the topology of pointwise convergence. The Schlicht-ing completion is a tdlc group. Notice moreover that if Γ is countable, then Sym(Γ / Λ) is a Polish group, so that Γ // Λ is Polish as well. Thus the Schlichtingcompletion of any countable group is a second countable tdlc group. For moreinformation on this construction we refer to [35, 37, 38].Clearly, every normal subgroup of Γ is commensurated. More generally, ev-ery subgroup that is commensurate to a normal subgroup is commensurated.Such subgroups are considered as “trivial examples” of commensurated sub-groups. They are characterized as follows. Lemma 5.1.
Let Γ be a group and Λ ≤ Γ a commensurated subgroup. Then Λ is commensurate to a normal subgroup of Γ if and only if the Schlichtingcompletion Γ // Λ is compact-by-discrete.Proof. If Γ // Λ is compact-by-discrete, i.e. if Γ // Λ has a compact open normalsubgroup, then the preimage of that subgroup in Γ is a normal subgroup thatis commensurate with Λ .Conversely, let N be a normal subgroup of Γ that is commensurate with Λ .Denote by ϕ : Γ → Γ // Λ the canonical homomorphism, whose image is dense.Then the closure ϕ ( N ) is a closed normal subgroup of Γ // Λ that is commensuratewith ϕ (Λ) . The latter is a compact open subgroup, so that ϕ ( N ) is compactand open as well. Thus Γ // Λ is compact-by-discrete. Lemma 5.2.
Let Γ be a finitely generated j.n.v.n. group. Let Λ ≤ Γ be a com-mensurated subgroup which is not commensurate to a normal subgroup. Thenthe homomorphism Γ → Γ // Λ is injective.If in addition Λ is not virtually contained in a normal subgroup of infiniteindex in Γ , then H = Γ // Λ admits a non-discrete, compactly generated, just-non-compact quotient group G , and the natural homomorphism ϕ : Γ → G isinjective with dense image.Proof. By Lemma 5.1, the Schlichting completion H = Γ // Λ is not compact-by-discrete. Since Γ is finitely generated, H is compactly generated. If thecanonical homomorphism Γ → H were not injective, then H would be virtuallynilpotent, hence compact-by-discrete in view of Proposition 3.5.28ince H is compactly generated, it has a closed normal subgroup N suchthat the quotient G = H/N is just-non-compact (see [9, Proposition 5.2]). Wedenote by ϕ the composite homomorphism Γ → H → G . By construction ϕ has dense image.Assume now that Λ is not virtually contained in a normal subgroup of infiniteindex in Γ . We claim that G is non-discrete. Indeed, otherwise N would beopen in H , and hence its pre-image in Γ would be a normal subgroup of infiniteindex that contains a finite index subgroup of Λ .Finally that map ϕ is injective, because otherwise ϕ (Γ) would be virtuallynilpotent, and hence G would be virtually nilpotent as well. In view of Propo-sition 3.5, this implies that G is discrete, which is not the case. Proposition 5.3.
Let Γ be a j.n.v.n. group such that every normal subgroup of Γ is finitely generated. Let Λ ≤ Γ be a commensurated subgroup which is notcommensurate to a normal subgroup of Γ .Then H = Γ // Λ has a closed normal subgroup such that the quotient group G satisfies all the conclusions of Theorem 3.13. Moreover the natural homo-morphism ϕ : Γ → G is injective with dense image.Proof. By Lemma 5.1, the Schlichting completion H = Γ // Λ is not compact-by-discrete. Since every normal subgroup of Γ is finitely generated, it followsthat every open normal subgroup of H is compactly generated. In particular Res( H ) is compactly generated, by Corollary 3.8. Therefore, we may invokeTheorem 3.13 to build a monolithic quotient G of H with the required prop-erties. The composite map ϕ : Γ → H → G has dense image. If ϕ were notinjective, then G would be virtually nilpotent, hence compact-by-discrete, whichviolates the conclusions of Theorem 3.13. Theorem 1.1 from the introduction is a consequenceof the following more comprehensive statement.
Theorem 5.4.
Let Γ be a finitely generated j.n.v.n. group, and let X be acompact Γ -space such that the action of Γ on X is faithful, minimal and micro-supported. Assume moreover that at least one of the following conditions holds.(1) Γ has a commensurated subgroup Λ that is of infinite index and that is notvirtually contained in a normal subgroup of infinite index of Γ .(2) Every normal subgroup of Γ is finitely generated, and Γ has a commensu-rated subgroup Λ that is not commensurate to a normal subgroup.Then the following assertions hold. i) The action of Γ on X is an almost boundary and has compressible opensubsets. In particular Γ is monolithic, hence not residually finite.(ii) There exists a compact Γ -space Y with Y ∼ hp X , such that the Γ -actionon Y extends to a continuous H -action, where H = Γ // Λ , and the quotientgroup G = H/K of H by the kernel of that action is monolithic with a non-discrete, non-amenable, compactly generated monolith M . Furthermore M coincides with Res( G ) . In particular G/M is compact-by-discrete.(iii) If in addition the Γ -action on X is strongly just-infinite, then one can take Y = X in (ii).Proof. We form the Schlichting completion H = Γ // Λ , and we invoke Lemma 5.2or Proposition 5.3 depending on whether we are in situation (1) or (2). In eithercase, we find a quotient G of H and an embedding ϕ : Γ → G with dense imageand such that G is monolithic with a compactly generated, non-compact, non-discrete monolith M , and G/M is compact-by-discrete.Consider the space Y := S ϕ,G ( X ) constructed in Section 4.2. The group G satisfies all the assumptions of Proposition 4.8, so it follows that the G -action on Y is faithful. Now consider the Γ -action on E ϕ ( X ) . The extension E ϕ ( X ) → X is almost one-to-one by Proposition 4.2, and hence it is highly proximal. Sincethe Γ -action on X is micro-supported, Proposition 2.3 implies that the Γ -actionon E ϕ ( X ) is micro-supported. Since the Γ -action on Y is also faithful (as the G -action is), it follows that the Γ -action on Y is micro-supported by Lemma2.1. Hence the G -action on Y is faithful, and also micro-supported as the Γ -action already has this property. Since in addition G is monolithic with acompactly generated, non-compact and non-discrete monolith, we have shownthat all the assumptions of Theorem 3.17 are satisfied. The latter thereforeimplies that the action of G on Y is an almost boundary and has compressibleclopen subsets. Since Γ is dense in G , it follows that the action of Γ on Y isalso an almost boundary with compressible clopen subsets. Now the Γ -spaces X and Y are highly proximally equivalent by Theorem 2.5, and by Lemma 2.2 andProposition 2.3 the property of being an almost boundary with compressibleopen subsets is invariant under highly proximal equivalence. Hence the space X also has this property. That the group Γ is monolithic then follows from thedouble commutator lemma, see Proposition I in [12]. So we have shown that (i)holds.Since the action of G on Y is faithful and micro-supported, by the doublecommutator lemma again we see that for every non-trivial normal subgroup N ,there exists a non-empty clopen subset U ⊆ Y such that N ≥ Rist G ( U ) ′ . Inparticular N has a non-trivial intersection with the image of Γ . Therefore G/N is30irtually nilpotent. In particular, if N is closed, then G/N is residually discreteby Proposition 3.5, so that
Res( G ) ≤ N . Since G is not residually discrete, weinfer that the monolith M of G coincides with Res( G ) . In particular G/M iscompact-by-discrete in view of Proposition 3.7. That M is not amenable followsfrom Theorem 3.17. So (ii) is proved.Finally if in addition the Γ -action on X is strongly just-infinite, then byProposition 4.9 the action of Γ on X extends to an action of G , which is faithfulsince the action of G on Y is faithful by (ii). This shows (iii). Remark . In the setting of the above theorem, it is not true that all micro-supported actions of Γ will extend to the Schlichting completion; see Remark 6.7.So the assumption in (iii) that the action is strongly just-infinite is necessary.Theorem 5.4 has at least two possible directions of applications:a) If the action of Γ on X is known not to satisfy conclusion (i), then by thetheorem we deduce that Γ does not admit any commensurated subgroupas in (1) or (2). See §6.1 and §6.2 for illustrations.b) On the other hand, when the group Γ does admit commensurated sub-groups as in the assumptions, then conclusion (iii) of the theorem tells usthat the action of Γ on X extends to the associated Schlichting comple-tions (via a quotient satisfying additional properties). Examples wheresuch a situation happens are given in §6.1.4. In this section we apply the results of Sections 4 and 5 to several classes ofgroups admitting a micro-supported action. If Λ is a group acting on a compact space X , we denoteby F (Λ , X ) the associated topological full group. Recall that F (Λ , X ) is thegroup of homeomorphisms g of X such that for every x ∈ X there exist aneighbourhood U of x and an element γ ∈ Λ such that g ( y ) = γ ( y ) for every y ∈ U .Given a non-empty clopen subset U of X and elements γ , . . . , γ n ∈ Λ suchthat γ ( U ) , . . . , γ n ( U ) are pairwise disjoint, there is an injective group homomor-phism Sym( n ) → F (Λ , X ) , σ g σ , where g σ is defined by g σ ( x ) = γ σ ( i ) γ − i ( x ) if x ∈ γ i ( U ) , and g σ ( x ) = x if x / ∈ ∪ γ i ( U ) . The alternating full group A (Λ , X ) ,31ntroduced by Nekrashevych in [33], is the subgroup of F (Λ , X ) generated bythe images of the alternating groups Alt( n ) under all such homomorphisms.The following is [33, Theorem 4.1] (see also [30]). Theorem 6.1.
Let Λ y X be a minimal action on a Cantor space X .Thenevery non-trivial subgroup of F (Λ , X ) that is normalized by A (Λ , X ) contains A (Λ , X ) . In particular A (Λ , X ) is simple and is contained in every non-trivialnormal subgroup of F (Λ , X ) . Recall that an action Λ y X on a Cantor space X is expansive if thereexist a compatible metric d and δ > such that for every x = y ∈ X thereexists γ ∈ Λ such that d ( γ ( x ) , γ ( y )) ≥ δ . For the following, see Proposition 5.7and Theorem 5.10 in [33]. Theorem 6.2.
Let Λ be a finitely generated group, and Λ y X a minimaland expansive action on a Cantor space X . Then the group A (Λ , X ) is finitelygenerated. The goal of this paragraphis to prove Theorem 6.4. We will use the following lemma:
Lemma 6.3.
Let Λ y X be a minimal action on a Cantor space X , and let Γ = A (Λ , X ) . If x, y ∈ X are distinct, then (cid:10) Γ x , Γ y (cid:11) = Γ .Proof. Let H = (cid:10) Γ x , Γ y (cid:11) , and let O x be the Γ -orbit of x in X . We check that H contains Γ z for every z ∈ O x . That will imply that H contains the normalsubgroup generated by Γ x , and hence that H = Γ byecause Γ is simple byTheorem 6.1.So given z ∈ O x , we wish to show that H contains Γ z . Clearly we mayassume that z is distinct from x and y , and it is enough to show that there exists g ∈ H such that g ( x ) = z , because then Γ z = g Γ x g − ≤ H . If z ′ is a point in O x different from x, y, z , it is possible to find three disjoint clopen subsets U x , U z , U z ′ containing respectively x, z, z ′ and not contaning y , and elements γ , γ , γ ∈ Λ such that γ ( U x ) = U z and γ ( x ) = z , γ ( U z ) = U z ′ and γ ( U z ′ ) = U x . Thehomeomorphism g of X that coincides on U x , U z and U z ′ respectively with γ , γ and γ and which acts trivially outside U x ∪ U z ∪ U z ′ is an element of Γ bydefinition of the alternating full group, and by construction g acts trivially ona neighbourhood of y . Hence g belongs to H , and g ( x ) = z , so the statement isproved. Theorem 6.4.
Let Λ be a finitely generated group, and Λ y X a minimal andexpansive action on a Cantor space X such that A (Λ , X ) y X does not admit ny compressible open subset. Let Γ be a subgroup of F (Λ , X ) that contains A (Λ , X ) . Then every commensurated subgroup of Γ is either finite or contains A (Λ , X ) .Proof. We first treat the case
Γ = A (Λ , X ) . Recall that under the presentassumptions, Γ is a finitely generated simple group (Theorems 6.1 and 6.2),and the action of Γ on X is strongly just-infinite by Lemma 6.3. Hence all theassumptions of Theorem 5.4 are verified. Hence according to part (i) of thistheorem, if Γ has a commensurated subgroup that is of infinite index and notvirtually contained in a normal subgroup of infinite index, then Γ y X admitsa compressible open subset. By our assumption this is not the case. So Γ hasno commensurated subgroup as above, and since Γ is simple this is equivalentto saying that every commensurated subgroup of Γ is either finite or equal to Γ . We now consider an arbitrary subgroup Γ of F (Λ , X ) that contains A (Λ , X ) .Suppose that Σ is a commensurated subgroup of Γ that does not contain A (Λ , X ) . Then Σ ′ = Σ ∩ A (Λ , X ) is a proper commensurated subgroup of A (Λ , X ) , and hence is finite according to the previous paragraph. Thereforesince A (Λ , X ) is normal in Γ , we infer that every element of A (Λ , X ) centralizesa finite index subgroup of Σ . Since A (Λ , X ) is a finitely generated group, itfollows that the entire A (Λ , X ) centralizes a finite index subgroup of Σ . Since A (Λ , X ) has trivial centralizer in F (Λ , X ) by Theorem 6.1, the subgroup Σ mustbe finite, as desired.Note that Theorem 1.2 and Corollary 1.3 follow from Theorem 6.4. In thefollowing example we illustrate that these results fail if we remove one of theassumptions. Example . Let ϕ denote the odometer on X := Z p , i.e. the homeomorphismof Z p defined by x x + 1 , and write Λ = h ϕ i . The topological full group F (Λ , X ) is considered in details in [21, Example 4.6]. It is an infinite group suchthat every finitely generated subgroup is virtually abelian.The action of Λ on X is minimal but not expansive. If T p is the p -regularrooted tree naturally associated to Z p (the coset tree associated to the se-quence of subgroups ( p Z p , p Z p , . . . ) ), then Aut( T p ) is naturally a subgroup of Homeo( X ) . The subgroup Σ := Aut( T p ) ∩ F (Λ , X ) of F (Λ , X ) is easily seen tobe infinite, of infinite index, and commensurated in F (Λ , X ) ; and Σ ∩ A (Λ , X ) is also infinite, of infinite index, and commensurated in A (Λ , X ) . Hence thisexample shows that the expansivity of the action of Λ on X in Theorem 1.2 andCorollary 1.3 is an essential assumption.Now remark that the action of F (Λ , X ) is minimal and expansive, but thegroup F (Λ , X ) is not finitely generated. Since F (Λ , X ) is equal to its own33opological full group, the previous paragraph also shows that Theorem 1.2 andCorollary 1.3 also fail without the finite generation assumption on the originalacting group. This paragraphdeals with the situation where there exist non-discrete tdlc groups into whichthe group A (Λ , X ) embeds densely. Examples of such groups are the Higman-Thompson groups, or topological full group associated with a one sided shift offinite type; see Section 6.1.4. Theorem 6.6.
Let Λ be a finitely generated group, and Λ y X a minimal andexpansive action on a Cantor space X . Assume that G is a tdlc group intowhich A (Λ , X ) embeds as a dense subgroup. Then the action of A (Λ , X ) on X extends to an action of G on X .Proof. Clearly we may assume that G is not discrete. The group A (Λ , X ) isfinitely generated by Theorem 6.2, so G is compactly generated. Moreover A (Λ , X ) is simple by Theorem 6.1, so G does not have any discrete quotient.Hence by Proposition 3.4 G has a topologically simple quotient Q , and A (Λ , X ) embeds densely in Q . Since the action of A (Λ , X ) on X is strongly just-infiniteby Lemma 6.3, we can apply Corollary 4.10, which says that the action of A (Λ , X ) extends to the group Q . Since Q is a quotient of G , in particular wehave shown that the action extends to an action of G . Wedenote by V d,k the Higman-Thompson group with parameters d ≥ and k ≥ acting on the Cantor space X d,k = { , . . . , k } · { , . . . , d } N by prefix replacement.We refer to [22] for a precise definition of these groups. Alternatively, V d,k canbe defined as the topological full group of a certain one sided shift of finite type[31]. In particular V d,k is equal to its own topological full group.The group V d,k admits an embedding with dense image in a non-discretetdlc group, namely the group AAut( T d,k ) of almost-automorphisms of the quasi-regular rooted tree T d,k [7]. In particular V d,k admits an infinite and infinite indexcommensurated subgroup. This fact has been recently generalized by Lederle,who showed that more generally every topological full group associated with aone sided shift of finite type admits infinitely many pairwise non-commensuratecommensurated subgroups [28].The group V d,k is well-known to be finitely generated and simple, and it iseasy to see that its action on the Cantor X d,k is strongly just-infinite. Henceit follows that V d,k satisfies all the assumptions of Theorem 5.4(iii), so that for34very dense embedding of V d,k into a tdlc group G , the action of V d,k on X d,k extends to G . Remark . In the above situation, it is not true that every minimal and micro-supported action of V d,k extends to G , and hence the assumption in Theorem5.4(iii) that the action is strongly just infinite is necessary. For example in thecase of G = AAut( T d, ) mentionned above (into which the group V d, embedsdensely), the group G admits a unique minimal and micro-supported action.This follows from the fact that G and the group Aut( T d +1 ) of automorphisms ofthe non-rooted regular tree of degree d +1 have isomorphic open subgroups (andhence have the same centralizer lattice) together with the fact that Aut( T d +1 ) has a unique minimal and micro-supported action [12, Theorem B.2]. So itfollows that the action of V d, on X d, is actually the unique minimal and micro-supported action that extends to G . Let T be a locally finite rooted tree, and Γ a group ofautomorphisms of T . We denote by ∂T the boundary of T . Given a vertex v ∈ T , we denote by T v the subtree of T that is below v , and by ∂T v theassociated clopen subset in ∂T . Note that the Γ -action on T extends to anaction by homeomorphisms on ∂T . We will denote by Rist Γ ( v ) := Rist Γ ( ∂T v ) the rigid stabilizer of v . For n ≥ , we also denote by Rist Γ ( n ) the rigid stabilizerof level n , that is the subgroup generated by Rist Γ ( v ) when v ranges over verticesof level n . Recall that the action of Γ is branch if Γ acts transitively on eachlevel of the tree (or, equivalently, if Γ acts minimally on ∂T ) and if Rist Γ ( n ) hasfinite index in Γ for all n ≥ .We will invoke the following result of Grigorchuk [20]. Theorem 6.8.
Let Γ ≤ Aut( T ) be a branch group, and N a non-trivial normalsubgroup of Γ . Then there exists a level n ≥ such that Rist Γ ( n ) ′ ≤ N . Inparticular every proper quotient of Γ is virtually abelian. We now give the proof of Theorem 1.6 by applying Theorem 5.4.
Proof of Theorem 1.6.
The group Γ is j.n.v.n. by Theorem 6.8. If T is a rootedtree on which Γ admits a faitful branch action, then the action of Γ on X = ∂T isfaithful, minimal and micro-supported. Since in addition every normal subgroupof Γ is finitely generated by Corollary A.5 of the appendix, it follows frompart (2) of Theorem 5.4 that if Γ had a commensurated subgroup that is notcommensurate to a normal subgroup, then conclusion (i) of the theorem wouldhold; which is clearly not the case here. So every commensurated subgroup of Γ is commensurate to a normal subgroup, and the statement is proved.35f Γ is a finitely generated just-infinite branch group, Theorem 1.6 says thatevery commensurated subgroup of Γ is finite or of finite index. In this specialcase this result has been proved by different means by Wesolek in [41]. HenceTheorem 1.6 extends Wesolek’s result to arbitrary finitely generated branchgroups. We denote by
Homeo( S ) the homeo-morphism group of the circle S , and by Homeo + ( S ) the subgroup of indextwo consisting of orientation preserving homeomorphisms. Given a subgroup Γ ≤ Homeo( S ) , we denote by Aut Γ ( S ) the centralizer of Γ in Homeo + ( S ) .Recall the following well-known classification: either Γ has a finite orbit in S ; or Γ admits an exceptional minimal set (i.e. there exists a unique closed non-emptyminimal Γ -invariant subset K ⊂ S , and K is homeomorphic to a Cantor set);or Γ acts minimally on S . See proposition 5.6 in [15]. Moreover for minimalactions we have the following result (see §5.2 in [15]). Theorem 6.9.
Assume that Γ ≤ Homeo + ( S ) acts minimally on S . Then ei-ther Aut Γ ( S ) is infinite and Γ is conjugate to a group of rotations, or Aut Γ ( S ) is a finite cyclic group, and the action of Γ on the topological circle Aut Γ ( S ) \ S is proximal. We will use the following lemma.
Lemma 6.10.
Let Γ be a subgroup of Homeo + ( S ) that is minimal and not con-jugated to a group of rotations, and let Λ be an infinite commensurated subgroupof Γ . Then the action of Λ on S is minimal.Proof. We have to show that Λ cannot have a finite orbit or an exceptionalminimal set. Upon passing to the action on Aut Γ ( S ) \ S , by Theorem 6.9 wemay assume that the action of Γ on S is proximal. Note that here this isequivalent to saying that every proper closed subset is compressible.Suppose that Λ has a finite orbit in S . Then upon passing to a finite indexsubgroup, we may assume that the set F of Λ -fixed points is non-empty. Notethat F is not the entire circle since Λ is infinite. Consider the action of Λ ona Λ -invariant open interval in the complement of F . By considering powers ofa suitable element, we see that there exists a non-empty open interval I suchthat every finite index subgroup of Λ contains an element λ such that λ ( I ) and I are disjoint. Fix such an interval I . By minimality and proximality of the Γ -action, there is γ ∈ Γ such that γ ( F ) ⊂ I . It follows that Λ ∩ γ Λ γ − is afinite index subgroup of Λ that fixes γ ( F ) ⊂ I , and we obtain a contradictionwith the definition of I . 36uppose now that Λ has an exceptionel minimal set K . Again by minimalityand proximality, we can find γ ∈ Γ such that γ ( K ) lies in a connected component J of the complement of K in S . Then the subgroup Λ ∩ γ Λ γ − stabilizes K and γ ( K ) . Hence Λ ∩ γ Λ γ − stabilizes J , and therefore cannot be of finite indexin Γ . Contradiction.Whenever Γ is a subgroup of Homeo( S ) , we denote by Γ the subgroup of Γ generated by the Γ x , x ∈ S . Equivalently, Γ is the subgroup of Γ generatedby the elements that fix pointwise an open interval in S . Proposition 6.11.
Let Γ be a subgroup of Homeo + ( S ) that is minimal andmicro-supported. Then Γ is monolithic, with monolith M = [Γ , Γ ] , and M issimple.Proof. Since the action of Γ on the circle is micro-supported, the group Aut Γ ( S ) must be trivial, and hence the action of Γ is proximal by Theorem 6.9. Sincehere proximality is equivalent to the fact that every proper closed subset is com-pressible, the statement then follows from a general result about such actions,see Proposition 4.6 in [25].We are now ready to prove Theorem 1.7 from the introduction. Proof of Theorem 1.7.
First observe that since the action of Γ on the circle ismicro-supported, the group Aut Γ ( S ) must be trivial, and hence the action of Γ is proximal by Theorem 6.9.Suppose that ϕ : Γ → G is a dense embedding of Γ into a tdlc group G . Weaim to show that G is discrete. Upon passing to a subgroup of index at mosttwo, we may assume that Γ acts on S by preserving the orientation. We argueby contradiction and assume that G is not discrete. Let U be a compact opensubgroup of G , and Λ = ϕ − ( U ) . The subgroup U is an infinite commensuratedsubgroup of Γ , and hence acts minimally on the circle by Lemma 6.10. Considerthe space E ϕ ( S ) and S ϕ,G ( S ) constructed in Section 4.2. Since the extension E ϕ ( S ) → S is almost one-to-one by Proposition 4.2, Λ also acts minimally on E ϕ ( S ) , and hence also on S ϕ,G ( S ) . Therefore S ϕ,G ( S ) is a compact G -spaceon which every compact open subgroup of G acts minimally. This is possibleonly if S ϕ,G ( S ) is a point, so S ϕ,G ( S ) = { N } for some closed normal subgroup N of G . By definition of S ϕ,G ( S ) , it follows that there is a dense subset X ϕ ⊆ S such that ϕ (Γ x ) = N for every x ∈ X ϕ . Now if z ∈ S is arbitrary and γ is anelement of Γ z , then by density we can find x ∈ X ϕ such that γ ∈ Γ z . Henceit follows that N contains ϕ (Γ ) , and hence N = ϕ (Γ ) . Since Γ has finiteindex in Γ by assumption, it follows that N has finite index in G , and that N is compactly generated. According to Lemma 4.3 we can find an open interval37 such that ϕ (Rist Γ ( I )) lies in B ( G ) . By proximality of the Γ -action and since B ( G ) is a normal subgroup of G , it follows that ϕ (Rist Γ ( J )) lies in B ( G ) forevery open interval J , ie ϕ (Γ ) ≤ B ( G ) . Since in addition B ( G ) is a closedsubgroup of G by Theorem 3.3 since G is compactly generated, it follows that B ( G ) is a finite index open subgroup of G . So by Theorem 3.2, we deduce that G admits a compact open normal subgroup K . Consider the subgroup ϕ − ( K ) .It is a non-trivial normal subgroup of Γ , and hence contains the monolith M of Γ , and M is simple by Proposition 6.11. Since no simple group can embed intoa profinite group, we have obtained a contradiction.Now let Λ be a commensurated subgroup of Γ , and assume for a contradic-tion that Λ is not commensurate to a normal subgroup of Γ . Since Γ has finiteindex in Γ by assumption, every proper quotient of Γ is virtually abelian byProposition 6.11. Hence by Lemma 5.2 the homomorphism Γ → Γ // Λ is injec-tive. But we have seen in by the previous paragraph that this implies that Γ // Λ is discrete, so we have obtained a contradiction. So Λ must be commensurateto a normal subgroup of Γ , and the statement is proved. Example . Fix two integers ℓ, k ≥ . Consider some integers n i ≥ , andwrite n = ( n , . . . , n k ) . Let P be the multiplicative group of R + > generated by n , . . . , n k , and we denote by A = Z [1 /n , . . . , /n k ] the ring of n -adic rationals,ie rational numbers whose denominator is in P . Denote by T ( ℓ, A, P ) the groupof piecewise linear homeomorphisms of R /ℓ Z with finitely many breakpoints,all in A , with slopes in P , and which preserve the n -adic rationals. In case ℓ = k = 1 and n = 2 , the group T (1 , Z [1 / , Z ) is Thompson’s group T .If we write Γ = T ( ℓ, A, P ) , the quotient Γ / Γ is finite, and was describedin [39] (see Theorem 5.2 and Lemma 5.4 there). Hence Theorem 1.7 applies tothis family of groups. In the case ℓ = k = 1 and n = 2 , we recover the factfrom [27] that every proper commensurated subgroup of Thompson’s group T is finite. In this final section, weapply the results of this paper to the situation where ϕ : H → G is a continuoushomomorphisms with dense image and G, H are both non-discrete tdlc groups.General results on dense embeddings into such topologically simple groups havebeen established in [10]. We start by recalling some terminology from loc. cit.Following [10], we say that a tdlc group G is expansive if there existsa compact open subgroup U ≤ G such that T g ∈ G gU g − = 1 . When G iscompactly generated, this is equivalent to asking that G admits a Cayley-Abelsgraph on which it acts faithfully. We say that G is regionally expansive if G contains a compactly generated open subgroup that is expansive. Notice that38ny such group G is first countable. We will also use the following terminologyfrom [10]. Definition 6.13.
We say that a tdlc group G is robustly monolithic if G is monolithic, and its monolith is non-discrete, regionally expansive and topo-logically simple. The class of robustly monolithic tdlc groups is denoted by R . By definition a group in R is necessarily non-discrete. It turns out that agroup in R is itself regionally expansive [10, Proposition 5.1.2], and hence inparticular first countable.Notice the inclusion S td ⊂ R , where S td is the class of compactly generatedtdlc groups that are non-discrete and topologically simple. The main motivationto introduce the class R is provided by [10, Theorem 1.1.2], which ensures theclass R is stable under taking dense locally compact subgroups. More precisely,given a continuous injective homomorphism of tdlc groups ϕ : H → G with denseimage such that H is non-discrete and G ∈ R , then H ∈ R . In particular, everynon-discrete dense locally compact subgroup of a group in S td belongs to R (but it may fail to be in S td ). Ω G as a maximal highly proximal extension. First recall the fol-lowing result [10, Theorem 1.2.5].
Theorem 6.14.
Every group G ∈ R is [A]-semisimple. Thus the centralizer lattice LC ( G ) is a Boolean algebra, and its Stone space Ω G satisfies the universal property from Theorem 3.15. The following resultrefines this universal property by providing additional information on the factormaps between Ω G and the faithful, micro-supported, compact G -spaces. Thenotions of maximal highly proximal extension and highly proximal equivalencehave been defined in Section 2. Theorem 6.15.
Let G ∈ R . Then all faithful, micro-supported, totally discon-nected compact G -spaces are minimal and highly proximally equivalent. Theircommon maximal highly proximal extension is Ω G . The proof requires the following result, which is of independent interest.
Proposition 6.16.
Let G be a tdlc group that is regionally expansive, [A]-semisimple and monolithic. Then for extension π : Y → X between faithful,micro-supported, totally disconnected compact G -spaces, there exists a point x ∈ X such that π − ( x ) is compressible. In particular if X, Y are minimal then π is highly proximal. roof. Let LC ( G ) be the centralizer lattice. In view of Theorem 3.15, we mayidentify X and Y with the Stone spaces of G -invariant Boolean subalgebras A X ⊆ A Y of LC ( G ) .By [10, Corollary 4.3.7], the group G has a compactly generated open sub-group O that is monolithic. Since O is open and G is [A]-semisimple, it followsthat O is [A]-semisimple. Moreover, we may find a compact open subgroup U of O that does not contain Mon( O ) . It then follows that T g ∈ O gU g − = 1 .Observe also that LC ( G ) = LC ( O ) since O is open.We may therefore invoke [12, Theorem 6.19]. This ensures that there exists α ∈ A X with α > such that for any β ∈ LC ( G ) , there exists g ∈ O with gα ≤ β .Let now x be a point of X contained in the clopen α . Let F x be theultrafilter on A X representing x . We set π ∗ ( F x ) = { γ ∈ A Y | ∃ α ∈ F x , α ≤ γ } . Notice that π ∗ ( F x ) is the filter on A Y which corresponds to the fiber π − ( x ) .Given any β > in A Y , we have seen that there exists g ∈ O with gα ≤ β .Therefore g − β ∈ π ∗ ( F x ) . This implies that the fiber π − ( gx ) = gπ − ( x ) iscontained in the clopen subset β , and hence this fiber is compressible. If inaddition X is minimal then this implies that all fibers are compressible, and π is highly proximal. Proof of Theorem 6.15.
By [10, Theorem 1.2.6], the G -action on Ω G is minimal(and strongly proximal). By Theorem 3.15, every totally disconnected compact G -space X on which the G -action is faithful and micro-supported is a G -factorof Ω G . In particular the G -action on X is minimal. By Proposition 6.16, any G -map Ω G → X is highly proximal. Moreover, given any highly proximalextension π : Y → X , the G -action on Y is micro-supported by Proposition 2.3,and hence by Theorem 3.15 there is a factor G -map Ω G → Y . This provesthat Ω G satisfies the property defining the maximal highly proximal extensionof X . Ω G with respect to dense embeddings.Theorem 6.17. Let
G, H ∈ R and ϕ : H → G be a continuous injective homo-morphism with dense image, and assume that G is compactly generated. Supposein addition that H/ Mon( H ) is compact, or that G/ Mon( G ) is compact.If Ω H is non-trivial, then the H -action on Ω G is faithful and micro-supported,and there exists a H -map Ω H → Ω G that is a highly proximal extension. roof. Since Ω H is non-trivial, it follows from [10, Proposition 7.2.3] that the H -action on Ω H is faithful. Hence by Theorem 3.15 it is micro-supported, andby [10, Theorem 1.2.6] it is minimal and strongly proximal. Since G is secondcountable, the conditions from Section 4.1 are thus fulfilled.Consider the spaces E ϕ (Ω H ) and Y := S ϕ,G (Ω H ) from Section 4.2. Recallthat E ϕ (Ω H ) is an almost one-to-one extension of Ω H by Proposition 4.2. Since Ω H admits no non-trivial highly proximal extension by Theorem 6.15, it followsthat the map from E ϕ (Ω H ) to Ω H is actually an isomorphism and that Y is afactor of Ω H .We claim that the H -action on Y is faithful. Suppose for a contradiction thatthe H -action on Y is not faithful. Then the G -action on Y is not faithful eitherand the subgroup N = ϕ (Mon( H )) of G acts trivially. If H/ Mon( H ) is compactthen G/N is compact since H is dense in G , and if G/ Mon( G ) is compact then G/N is also compact since N contains Mon( G ) . Hence in both situations the G -action on Y factors through a compact group. On the other hand, recallthat the H -action on Ω H is strongly proximal. Since strong proximality passesto factors, the H -action on Y is strongly proximal. Since the only stronglyproximal action of a compact group is the trivial action, it follows that Y istrivial. In case G/ Mon( G ) is compact then every non-trivial closed normalsubgroup of G is cocompact, so in particular compactly generated. Hence inthis situation we can apply Lemma 4.3, and we derive a contradiction as theassumption G ∈ R implies that B ( G ) is trivial (otherwise by the same argumentas in the proof of Proposition 4.7, we would obtain a closed normal subgroup of G that is compact-by-discrete, which is impossible here). Hence it only remainsto obtain a contradiction when H/ Mon( H ) is compact. Let J ≤ H be a non-trivial closed subgroup whose normalizer N H ( J ) is of finite index. Then N H ( J ) contains an open normal subgroup of H , and hence we have Mon( H ) ≤ N H ( J ) .Thus J and Mon( H ) normalize each other. In particular J ∩ Mon( H ) is aclosed normal subgroup Mon( H ) . Since the latter is topologically simple, witha trivial centralizer in H , it follows that Mon( H ) is contained in J , and hence J is cocompact in H . Hence we may invoke Proposition 4.4, and again since B ( G ) is trivial we have reached a contradiction. So the H -action on Y is faithful, andhence it follows from Lemma 2.1 that the H -action on Y is also micro-supported.In particular, the G -action on Y is micro-supported as well.We claim that the action of G on Y is also faithful. According to the previousparagraph, we are in position to apply Proposition 3.16, which provides a G -equivariant order-preserving injective map f : A → LC ( G ) , where A is theBoolean algebra of clopen subsets of Y . If the G -action on Y were not faithful,then Mon( G ) would act trivially on Y , hence also on f ( A ) . On the other hand,by [10, Proposition 7.2.3], the only Mon( G ) -fixed points in LC ( G ) are the trivial41nes and ∞ , a contradiction. This proves the claim.We may now invoke Theorem 6.15, and deduce that there is G -map Ω G → Y that is a highly proximal extension. Since the H -action on Y is micro-supported, by Proposition 2.3 the H -action on Ω G is also micro-supported, andthe statement follows by applying again Theorem 6.15.Recall that if K is a compact subgroup of a tdlc group G , the commensurator Comm G ( K ) carries a unique tdlc group topology such that the inclusion map K → Comm G ( K ) is continuous and open (see for instance [12, Lemma 5.13]).We denote by G ( K ) the group Comm G ( K ) endowed with that topology.Combining the above theorem with a result from [10], we derive the followingconsequence. Corollary 6.18.
Let G ∈ S td and K ≤ G be an infinite compact subgroup.Assume that one of the following conditions holds:(1) K is locally normal and Comm G ( K ) = G .(2) K is a pro- p Sylow subgroup of some compact open subgroup of G , forsome prime p .Then there exists a G ( K ) -map Ω G ( K ) → Ω G that is a highly proximal exten-sion.Proof. If (1) holds, we have
Comm G ( K ) = G as abstract groups. If (2) holds,we know from [34, Theorem 1.2] that Comm G ( K ) is dense in G . In either caseby [10, Theorem 1.1.2], we have G ( K ) ∈ R .Observe that LC ( K ) = LC ( G ( K ) ) . Clearly there is nothing to prove if LC ( G ) and LC ( G ( K ) ) are both trivial. Assume first that LC ( G ) is non-trivial. If (1)holds, then the action of G ( K ) on Ω G is faithful and micro-supported as G ( K ) = G as abstract groups. If (2) holds then the same is true according to [10,Theorem 8.4.1]. Hence in either case the statement follows from Theorem 6.15.Conversely if LC ( K ) is non-trivial, then we are in position to apply Theorem6.17 since G lies in S td , which gives the statement.It follows in particular that if G ∈ S td is of atomic type in the terminologyof [12, Theorem F], then every locally normal subgroup K of G has a trivialcentralizer lattice. Indeed, by [12, Theorem F], if the group G is of atomictype, then its action on the structure lattice LN ( G ) is trivial. This implies thatevery compact locally normal subgroup K is commensurated in G , and that thecentralizer lattice LC ( G ) is trivial (since otherwise G would act non-trivially,hence faithfully, on LC ( G ) by [12, Theorem J]; its action on LN ( G ) is thus afortiori faithful). Hence by the above theorem LC ( K ) is trivial.42 Normal subgroups of finitely generated branchgroups are finitely generated — by DominikFrancoeur
Our goal is to prove that every normal subgroup of a finitely generated branchgroup is finitely generated, see Corollary A.5. We begin with a lemma, whichis a variation of what is sometimes known as the double commutator lemma .This result has appeared in various forms and degrees of generality over theyears. It states that for every non-trivial normal subgroup N of a group G ofhomeomorphisms of a Hausdorff space X , there exists a non-empty open subset U ⊆ X such that Rist ′ G ( U ) ≤ N , where Rist ′ G ( U ) is the derived subgroup ofthe rigid stabiliser Rist G ( U ) . In the following lemma, we prove that under somestronger conditions on the action of G , we can find in N not only Rist ′ G ( U ) , buta finitely generated subgroup containing it. As the proof involves taking threecommutators instead of the usual two, it was suggested to us that we name itthe triple commutator lemma . Lemma A.1 (Triple commutator lemma) . Let G be a group with a micro-supported faithful action by homeomorphisms on a Hausdorff space X . Supposethat there exists a base B for the topology on X such that for every open subset U ∈ B , the rigid stabiliser Rist G ( U ) is finitely generated. Then, for every non-trivial normal subgroup N E G , there exists a non-empty open subset U ∈ B and a finitely generated subgroup H ≤ N such that Rist ′ G ( U ) ≤ H ≤ N .Proof. Let g ∈ N be a non-trivial element of N . Since the action of G on X is faithful, there exists an open subset V ⊆ X such that gV = V . Since X isHausdorff, replacing V by a smaller open subset if necessary, we can assumethat gV ∩ V = ∅ .As the action of G on X is micro-supported, there must exist an element t ∈ Rist G ( gV ) and an open subset W ⊆ V such that tW ∩ W = ∅ . Let uschoose a non-empty basic open subset U ∈ B contained in g − W . We then havethat U, gU and tgU are three pairwise disjoint open subsets of X .By our hypothesis, Rist G ( U ) is finitely generated. Let { r , . . . , r n } be afinite symmetric generating set for Rist G ( U ) , and let H ≤ G be the subgroupgenerated by (cid:8) [ g, r − ] , . . . , [ g, r − n ] (cid:9) ∪ (cid:8) [ tgt − , r − ] , . . . , [ tgt − , r − n ] (cid:9) . The fact that N is normal implies directly that H ≤ N . We will now show that Rist ′ G ( U ) ≤ H . 43et r ∈ Rist G ( U ) be an arbitrary element. Since { r , . . . , r n } is a sym-metric generating set for Rist G ( U ) , there exist ≤ i , . . . , i k ≤ n such that r = r i . . . r i k . For ≤ i ≤ n , we have [ g, r − i ] = ( gr − i g − ) r i , with r i ∈ Rist G ( U ) and gr − i g − ∈ Rist G ( gU ) . Since U and gU are disjoint open subsets of X , theelements of Rist G ( U ) commute with the elements of Rist G ( gU ) . Thus, we have [ g, r − ] = gr − g − r = [ g, r − i ] . . . [ g, r − i k ] ∈ H. Using the same notation, we also have, for ≤ i ≤ n , [ tgt − , r − i ] = ( tgt − r − i tg − t − ) r i = (( tg ) r − i ( tg ) − ) r i , where the last equality comes from the fact that t and r i commute, since t ∈ Rist G ( gV ) and r i ∈ rist G ( U ) ≤ Rist G ( V ) , with V ∩ gV = ∅ . Note that we have ( tg ) r − i ( tg ) − ∈ Rist G ( tgU ) and r i ∈ Rist G ( U ) . Since U ∩ tgU = ∅ , elements of Rist G ( U ) and Rist G ( tgU ) commute. Therefore, as above, we have [ tg, r − ] = ( tg ) r − ( tg ) − r = [ tgt − , r − i ] . . . [ tgt − , r − i k ] ∈ H. It follows that for all r, s ∈ Rist G ( U ) , we have [[ g, r − ] , [ tg, s − ]] = [( gr − g − ) r, (( tg ) s − ( tg ) − ) s ] ∈ H. However, since gr − g − ∈ Rist G ( gU ) and ( tg ) s − ( tg ) − ∈ Rist G ( tgU ) , we get,using the fact that U, gU and tgU are pairwise disjoint sets, that gr − g − com-mutes with r , s and ( tg ) s − ( tg ) − , and that ( tg ) s − ( tg ) − commutes with r , s and gr − g − . It follows that [[ g, r − ] , [ tg, s − ]] = [ r, s ] , and so [ r, s ] ∈ H . The result follows immediately from the fact that Rist ′ G ( U ) is generated by elements of the form [ r, s ] with r, s ∈ Rist G ( U ) .Using this lemma, we can conclude that every normal subgroup of a finitelygenerated branch group is itself finitely generated. In fact, we will prove aslightly more general result. Let us first recall a few definitions. Definition A.2.
A group G is said to be Noetherian , or to satisfy the maxi-mum condition on subgroups, if every subgroup of G is finitely generated. Definition A.3.
Let T be a spherically homogeneous locally finite rooted tree T . A subgroup G ≤ Aut ( T ) of the group of automorphisms of T is said to bea weakly branch group if it acts transitively on each level of the tree and iffor all vertex v ∈ T , the rigid stabiliser Rist G ( v ) is non-trivial.44 heorem A.4. Let G be a finitely generated weakly branch group acting ona spherically homogeneous locally finite rooted tree T . If Rist G ( n ) is finitelygenerated and if G/ Rist ′ G ( n ) is Noetherian for all n ∈ N , then every normalsubgroup N E G is finitely generated.Proof. If N = 1 , then it is obviously finitely generated. Let us now assume that N = 1 .Since G is a weakly branch group, its action on the boundary of the rootedtree T , which is homeomorphic to the Cantor set, is micro-supported. If weassume that Rist G ( n ) is finitely generated for all n ∈ N , it then satisfies thehypotheses of Lemma A.1. Therefore, there must exist a vertex v ∈ T on level n ∈ N and a finitely generated subgroup H ≤ N such that Rist ′ G ( v ) ≤ H .Let { t , . . . , t k } be a transversal of St G ( v ) , so that Rist ′ G ( n ) = Q ki =1 Rist ′ G ( t i v ) .Then, the subgroup generated by S ki =1 t i Ht − i is a finitely generated subgroupof N , since H is finitely generated and N is normal, and it contains Rist ′ G ( n ) ,since it contains t i Rist ′ G ( v ) t − i = Rist ′ G ( t i v ) for all ≤ i ≤ k .Let us consider the quotient G/ Rist ′ G ( n ) . By assumption, every subgroup ofthis group is finitely generated. In particular, N/ Rist ′ G ( n ) is finitely generated.Since Rist ′ G ( n ) is contained in a finitely generated subgroup of N , we concludethat N itself must also be finitely generated. Corollary A.5.
Every normal subgroup of a finitely generated branch group isfinitely generated.Proof.
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