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

## TL;DR

This paper investigates the properties of alternating groups arising from topologically free actions of infinite groups on the Cantor set, revealing their vanishing -Betti numbers, stability, property Gamma, and conditions for simplicity and C*-simplicity.

## Contribution

It establishes new results on the stability, property Gamma, and inner amenability of these alternating groups, especially in the context of amenable groups, and introduces a topological approach to measure entropy invariance.

## Key findings

- All such alternating groups have vanishing -Betti numbers.
- For amenable , these groups are often simple, finitely generated, and C*-simple.
- Many examples are shown to be stable and possess property Gamma.

## 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$^*$-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.