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

Abstract

We prove that the alternating group of a topologically free action of a countably infinite group $\Gamma$ on the Cantor set has the property that all of its ${\ell }^2$-Betti numbers vanish and, in the case that $\Gamma$ 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 $\Gamma$ that there are many such alternating groups which are simple, finitely generated, and C${}^{\ast }$-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.

30 pages, to appear in Annales Scientifiques de l'ENS