A dichotomy for topological full groups

Abstract

Given a minimal action $\alpha$ of a countable group on the Cantor set, we show that the alternating full group $\mathsf{A}(\alpha )$ is non-amenable if and only if the topological full group $\mathsf{F}(\alpha )$ is $C^{\ast }$-simple. This implies, for instance, that the Elek-Monod example of non-amenable topological full group coming from a Cantor minimal ${\mathbb{Z}}^2$-system is $C^{\ast }$-simple.

6 pages. Fixed a couple of incorrections and added details to some of the proofs. Accepted in the Canadian Mathematical Bulletin