Non-free almost finite actions for locally finite-by-virtually $\mathbb{Z}$ groups

Abstract

In this paper, we study almost finiteness and almost finiteness in measure of non-free actions. Let $\alpha:G\curvearrowright X$ be a minimal action of a locally finite-by-virtually $\mathbb{Z}$ group $G$ on the Cantor set $X$. We prove that under certain assumptions, the action $\alpha$ is almost finite in measure if and only if $\alpha$ is essentially free. As an application, we obtain that any minimal topologically free action of a virtually $\mathbb{Z}$ group on an infinite compact metrizable space with the small boundary property is almost finite. This is the first general result, assuming only topological freeness, in this direction, and these lead to new results on uniform property $\Gamma$ and $\mathcal{Z}$-stability for their crossed product $C^*$-algebras. Some concrete examples of minimal topological free (but non-free) subshifts are provided.