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

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 an infinite compact metrizable space $X$. We prove $\alpha$ is almost finite in measure if and only if $\alpha$ is essentially free and $X$ has the small boundary property. 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 seems the first general result, assuming only topological freeness, in this direction. These also lead to new results on uniform property $\Gamma$ and $\mathcal{Z}$-stability for crossed product $C^*$-algebras.