Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups

Adams, Scot

doi:10.1016/0040-9383(94)90007-8

Cited by 84 publications

(156 citation statements)

References 5 publications

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“…(This equivalent description can be extracted from the results of [44], and is nearly explicit in Proposition 4.4.5(2) from [13].) For example, word-hyperbolic groups are amenable at infinity, see [1] and also Appendix B in [5] (or else Theorem 5.3.15 in [13]). It was shown that amenability at infinity is equivalent to a property of importance in operator algebra theory, exactness of a group (cf.…”

Section: Some Classes Of Groups To Look Atmentioning

confidence: 99%

Hyperlinear and Sofic Groups: A Brief Guide

2008

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Abstract. This is an introductory survey of the emerging theory of two new classes of (discrete, countable) groups, called hyperlinear and sofic groups. They can be characterized as subgroups of metric ultraproducts of families of, respectively, unitary groups U (n) and symmetric groups S n , n ∈ N. Hyperlinear groups come from theory of operator algebras (Connes' Embedding Problem), while sofic groups, introduced by Gromov, are motivated by a problem of symbolic dynamics (Gottschalk's Surjunctivity Conjecture). Open questions are numerous, in particular it is still unknown if every group is hyperlinear and/or sofic.

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Section: Some Classes Of Groups To Look Atmentioning

confidence: 99%

Hyperlinear and Sofic Groups: A Brief Guide

2008

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“…First consider µ = ge ∈ vG 1 . Then e ← − = v and [v, r(e)] = v. Covariance of (i A , i π ) gives i π (g)i A (χ Z(1e) ) = i A (χ Z(ge) )i π (g), and hence…”

Section: From Lemma 43 We Getmentioning

confidence: 99%

C⁎-algebras associated to graphs of groups

Brownlowe

Mundey

Pask

et al. 2017

Advances in Mathematics

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Abstract. To a large class of graphs of groups we associate a C * -algebra universal for generators and relations. We show that this C * -algebra is stably isomorphic to the crossed product induced from the action of the fundamental group of the graph of groups on the boundary of its Bass-Serre tree. We characterise when this action is minimal, and find a sufficient condition under which it is locally contractive. In the case of generalised Baumslag-Solitar graphs of groups (graphs of groups in which every group is infinite cyclic) we also characterise topological freeness of this action. We are then able to establish a dichotomy for simple C * -algebras associated to generalised BaumslagSolitar graphs of groups: they are either a Kirchberg algebra, or a stable Bunce-Deddens algebra.

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“…Classes of discrete groups known to be C * -exact include amenable groups [Lan73], word hyperbolic groups [Ada94,Ger98], and, by an argument of Connes, discrete subgroups of connected Lie groups [KW99]. The conjecture that all countable discrete groups are C * -exact has been disproven by recent work of Gromov, Guentner-Kaminker and Ozawa [Gro99,GK99,Oza00] which shows that there exists a countable (even geometrically finite) discrete group that is not C * -exact.…”

Section: Introductionmentioning

confidence: 99%

Exactness of one relator groups

Guentner

2001

Proc. Amer. Math. Soc.

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Abstract. A discrete group Γ is C * -exact if the reduced crossed product with Γ converts a short exact sequence of Γ-C * -algebras into a short exact sequence of C * -algebras. A one relator group is a discrete group Γ admitting a presentation Γ = X | R where X is a countable set and R is a single word over X. In this short paper we prove that all one relator discrete groups are C * -exact. Using the Bass-Serre theory we also prove that a countable discrete group Γ acting without inversion on a tree is C * -exact if the vertex stabilizers of the action are C * -exact.

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Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups

Cited by 84 publications

References 5 publications

Hyperlinear and Sofic Groups: A Brief Guide

Hyperlinear and Sofic Groups: A Brief Guide

C⁎-algebras associated to graphs of groups

Exactness of one relator groups

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