A dichotomy for topological full groups

TL;DR

This paper establishes a precise relationship between non-amenability and $C^*$-simplicity in topological full groups arising from minimal actions on the Cantor set, revealing a dichotomy that clarifies their algebraic structure.

Contribution

It proves that for minimal actions on the Cantor set, the alternating full group is non-amenable if and only if the topological full group is $C^*$-simple, providing a clear dichotomy.

Findings

The alternating full group is non-amenable iff the topological full group is $C^*$-simple.

The Elek-Monod example from a Cantor minimal $bZ^2$-system is $C^*$-simple.

Established a dichotomy linking algebraic properties of these 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 $C^*$-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 $C^*$-simple.