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

2019/02/11 by Kerr, David, Tucker-Drob, Robin · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1902.04131

Abstract

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 ℓ²-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 many such alternating groups which are simple, finitely generated, and C^*-simple. The device for establishing nonisomorphism among these examples is a topological version of Austin's result on the invariance of measure entropy under bounded orbit equivalence.

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