Growth conditions for topological freeness

Abstract

Given a finitely generated group $\Gamma$, a non-trivial element $g\in \Gamma$, and a minimal action $\Gamma \curvearrowright \mathcal{X}$ on a compact space $\mathcal{X}$, with amenable neighborhood stabilizers, we prove sufficient conditions in terms of various growth/decay functions for topological freeness of the action of $g$ on $\mathcal{X}$. We apply our results to the case of the Furstenberg boundary action of $\Gamma$, to conclude C*-simplicity under (sub-)rapid decay conditions. In this context, we also give a description of the support of stationary states on the reduced C*-algebra of $\Gamma$.

Several results generalized; concrete examples added