Almost finiteness and groups of dynamical origin

Abstract: We introduce the property of having good subgroups for actions of countable discrete groups on compact metrizable spaces, and show that it implies comparison when the acting group is amenable. As a consequence, free actions on finite-dimensional spaces of many notable amenable groups of dynamical origin are almost finite. For instance, this applies to topological full groups of Cantor minimal systems and the Basilica group. In particular, minimal such actions give rise to classifiable crossed products.