Nonfree almost finite actions for locally finite‐by‐virtually Z${\mathbb {Z}}$ groups

Abstract

AbstractIn this paper, we study almost finiteness and almost finiteness in measure of nonfree actions. Let be a minimal action of a locally finite‐by‐virtually group on the Cantor set . We prove that under certain assumptions, the action is almost finite in measure if and only if is essentially free. As an application, we obtain that any minimal topologically free action of a virtually 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 and ‐stability for their crossed product ‐algebras. Some concrete examples of minimal topological free (but nonfree) subshifts are provided.