Extensive amenability and an application to interval exchanges

Juschenko, Kate; Bon, Nicolás Matte; Monod, Nicolas

doi:10.1017/etds.2016.32

Cited by 29 publications

(64 citation statements)

References 24 publications

(30 reference statements)

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“…This result motivated the study of analytic properties of topological full groups. Amenability of other families of topological full groups was established in [JNdlS16,JMBMS17]. Recently Nekrashevych constructed étale groupoids whose topological full groups have intermediate growth [Nek16], giving the first examples of simple groups with this property.…”

Section: Piecewise Projective Homeomorphisms Of the Real Line Followmentioning

confidence: 99%

Subgroup dynamics and C*-simplicity of groups of homeomorphisms

Boudec¹,

Bon²

2018

Ann. Sci. École Norm. Sup.

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We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a C * -simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group V is C * -simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented C * -simple groups without free subgroups. We prove that a branch group is either amenable or C * -simple. We also prove the converse of a result of Haagerup and Olesen: if Thompson's group F is non-amenable, then Thompson's group T must be C * -simple. Our results further provide sufficient conditions on a group of homeomorphisms under which uniformly recurrent subgroups can be completely classified. This applies to Thompson's groups F , T and V , for which we also deduce rigidity results for their minimal actions on compact spaces.

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Section: Piecewise Projective Homeomorphisms Of the Real Line Followmentioning

confidence: 99%

Subgroup dynamics and C*-simplicity of groups of homeomorphisms

Boudec¹,

Bon²

2018

Ann. Sci. École Norm. Sup.

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“…These groups can be realized as topological full groups of minimal action of Z 2 on the Cantor set. Therefore, Theorem 1 in the combination with [14], [11] gives more examples of simple finitely generated infinite amenable groups, and thus by the result of Chou non-elementary amenable groups. Matui, [14], showed that…”

Section: Introductionmentioning

confidence: 91%

“…It was proved in [14], that the commutator subgroup of [[Z d ]] is simple. The topological full groups that correspond to interval exchange transformation group were studied in [11]. The authors prove that subgroups of rank equals to 2 are amenable.…”

Section: Introductionmentioning

confidence: 99%

On topological full groups of $\mathbb Z^d$-actions

2020

Self Cite

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“…A partial, positive result appears in Example 9.15. It would suffice, following the strategy in that example (see [70,Proposition 5.3]), to prove that the group of Z dwobbles W (Z d ) acts extensively amenably on Z d for all d ∈ N; at present (2017), this is known only for d ≤ 2, see Theorem 9.16.…”

Section: Problem 113 (Geoghegan) Is Thompson's Group F Amenable?mentioning

confidence: 99%

Amenability of Groups and G-Sets

Bartholdi

2018

Trends in Mathematics

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This text surveys classical and recent results in the field of amenability of groups, from a combinatorial standpoint. It has served as the support of courses at the University of Göttingen and theÉcole Normale Supérieure. The goals of the text are (1) to be as self-contained as possible, so as to serve as a good introduction for newcomers to the field; (2) to stress the use of combinatorial tools, in collaboration with functional analysis, probability etc., with discrete groups in focus; (3) to consider from the beginning the more general notion of amenable actions; (4) to describe recent classes of examples, and in particular groups acting on Cantor sets and topological full groups.(This last property is often called finite additivity, as opposed to the σ -additivity property enjoyed by measures, in which countable unions are allowed).It easily follows from the definition that m( / 0) = 0; that m(A) ≤ m(B) if A ⊆ B; and that m(A 1 · · · A k ) = m(A 1 ) + · · · + m(A k ) for pairwise disjoint A 1 , . . . , A k .We denote by M (X) the set of means on X, with the usual topology on a set of functions; namely, a sequence m n ∈ M (X) converges to m precisely if for every ε > 0 and every finite collection A 1 , . . . , A k ⊆ X we have |m n (A i ) − m(A i )| < ε for all i ∈ {1, . . . , k} and all n large enough.Observe that M is a covariant functor: if f : X → Y , then we have a natural mapIn particular, if a group G acts on X, then it also acts on M (X). For a right action · : X × G → X, we have a right action on M (X) given by (m · g)(A) = m(A · g −1 ) for all A ⊆ X. [103]). Let G be a group and let X G be a set on which G acts. The G-set X is amenable if there is a G-fixed element in M (X). Definition 2.2 (von NeumannA group G is amenable if all non-empty right G-sets are amenable.In other words, the G-set X is amenable if M (X) G = / 0, namely if there exists a mean m on X such that m(Ag) = m(A) for all g ∈ G and all A ⊆ X.

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Extensive amenability and an application to interval exchanges

Cited by 29 publications

References 24 publications

Subgroup dynamics and C*-simplicity of groups of homeomorphisms

Subgroup dynamics and C*-simplicity of groups of homeomorphisms

On topological full groups of $\mathbb Z^d$-actions

Amenability of Groups and G-Sets

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