A dichotomy for topological full groups

Given a minimal action $α$ of a countable group on the Cantor set, we show that the alternating full group $\mathsf{A}(α)$ is non-amenable if and only if the topological full group $\mathsf{F}(α)$ 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.

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