Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity

Shalom, Yehuda; Willis, George A.

doi:10.1007/s00039-013-0236-5

Cited by 56 publications

(56 citation statements)

References 53 publications

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“…We would like to point out that Schlichting completions are sometimes called relative profinite completions [15,31], but we choose not to use this terminology in order to avoid confusion with the notion of localised profinite completion appearing in [28]. Although we will not use this terminology, we also note that a group together with a commensurated subgroup is sometimes called a Hecke pair.…”

Section: Compact Presentability Of Schlichting Completionsmentioning

confidence: 99%

Compact presentability of tree almost automorphism groups

Boudec¹

2017

Annales De l'Institut Fourier

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We establish compact presentability, i.e. the locally compact version of finite presentability, for an infinite family of tree almost automorphism groups. Examples covered by our results include Neretin's group of spheromorphisms, as well as the topologically simple group containing the profinite completion of the Grigorchuk group constructed by Barnea, Ershov and Weigel.We additionally obtain an upper bound on the Dehn function of these groups in terms of the Dehn function of an embedded Higman-Thompson group. This, combined with a result of Guba, implies that the Dehn function of the Neretin group of the regular trivalent tree is polynomially bounded.

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Section: Compact Presentability Of Schlichting Completionsmentioning

confidence: 99%

Compact presentability of tree almost automorphism groups

Boudec¹

2017

Annales De l'Institut Fourier

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“…For some other groups, all almost normal subgroups are either finite of have a finite index. This is for instance the case of SL n (Z) with n ≥ 3 (see [42]). For such groups , the only possible faithful t.d.l.c.…”

Section: The Schlichting Completion Associated With a Fso Action mentioning

confidence: 99%

Amenable actions preserving a locally finite metric

Anantharaman-Delaroche

2018

Expositiones Mathematicae

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The class A of countable groups that admit a faithful, transitive, amenable -in the sense that there is an invariant mean -action on a set has been widely investigated in the past. In this paper, we no longer require the action to be transitive, but we ask for it to preserve a locally finite metric (and still to be faithful and amenable). The groups having such actions are those that embed into a totally disconnected amenable locally compact group. Then we focus on the subclass A 1 of groups for which the actions are moreover transitive. This class is strictly contained into A and includes non-amenable groups. An important particular case of actions preserving a locally finite metric is given by actions by automorphisms of locally finite connected graphs. We take this opportunity, in our partly expository paper, to review some nice results about amenable actions in this setting.

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“…In the preprint [25], Y. Shalom and G. A. Willis use the term "commensurated" and we follow that terminology. They show that for n > 2, all commensurated subgroups of SL n .Z/, are finite or of finite index.…”

Section: Introductionmentioning

confidence: 99%

“…They also show SL n .ZOE1=p/ satisfies Margulis' theorem but its subgroup SL n .Z/ is commensurated. Still, under certain hypotheses, there are commensurated versions of Margulis' theorem (see [25]). …”

Section: Introductionmentioning

confidence: 99%

Commensurated subgroups and ends of groups

Conner

2013

Journal of Group Theory

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Abstract. If G is a group, then subgroups A and B are commensurable if A \ B has finite index in both A and B. The commensurator of A in G, denoted by Comm G .A/, is ¹g 2 G j .gAg 1 / \ A has finite index in both A and gAgIt is straightforward to check thatWe develop geometric versions of commensurators in finitely generated groups. In particular, g 2 Comm G .A/ iff the Hausdorff distance between A and gA is finite. We show a commensurated subgroup of a group is the kernel of a certain map, and a subgroup of a finitely generated group is commensurated iff a Schreier (left) coset graph is locally finite. The ends of this coset graph correspond to the filtered ends of the pair .G; A/. This last equivalence is particularly useful for deriving asymptotic results for finitely generated groups. Our primary goals in this paper are to develop and compare the basic theory of commensurated subgroups to that of normal subgroups, and to initiate the development of the asymptotic theory of commensurated subgroups.

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Commensurated Subgroups of Arithmetic Groups, Totally Disconnected Groups and Adelic Rigidity

Cited by 56 publications

References 53 publications

Compact presentability of tree almost automorphism groups

Compact presentability of tree almost automorphism groups

Amenable actions preserving a locally finite metric

Commensurated subgroups and ends of groups

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