A dichotomy for topological full groups

Abstract

AbstractGiven 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.