A Tits alternative for topological full groups

Szőke, Nóra Gabriella

doi:10.48550/arxiv.1808.09882

Cited by 1 publication

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“…It has also recently been shown by Szőke that every infinite finitely generated group which is not virtually cyclic admits a free minimal expansive action on the Cantor set whose topological full group contains F 2 [45].…”

Section: Introductionmentioning

confidence: 98%

“…A celebrated theorem of Juschenko and Monod says that this is the case for all minimal Z-actions on the Cantor set [23]. It is also the case if the action is free, minimal, and equicontinuous (Theorem 1.3 of [12]) or if the action is free and minimal and the acting group is virtually cyclic [45]. On the other hand, the alternating group of the shift action Z {1, .…”

Section: Introductionmentioning

confidence: 99%

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Dynamical alternating groups, stability, property Gamma, and inner amenability

Kerr¹,

Tucker-Drob²

2019

Preprint

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We prove that the alternating group of a topologically free action of a countably infinite group Γ on the Cantor set has the property that all of its ℓ 2 -Betti numbers vanish and, in the case that Γ is amenable, is stable in the sense of Jones and Schmidt and has property Gamma (and in particular is inner amenable). We show moreover in the realm of amenable Γ that there are large classes of such alternating groups which are simple, finitely generated, and nonamenable, and in many cases also C * -simple. Among the tools used in constructing some of these classes is a topological version of Austin's result on the invariance of measure entropy under bounded orbit equivalence.

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Section: Introductionmentioning

confidence: 98%