Group extensions preserve almost finiteness

Abstract: We show that a free action $G \curvearrowright X$ is almost finite if its restriction to some infinite normal subgroup of $G$ is almost finite. Consider the class of groups which contains all infinite groups of locally subexponential growth and is closed under taking inductive limits and extensions on the right by any amenable group. It follows that all free actions of a group from this class on finite-dimensional spaces are almost finite and therefore that minimal such actions give rise to classifiable crossed products. In particular, that gives a much easier proof for the recent result of Kerr and the author on elementary amenable groups.