Clopen type semigroups of actions on $0$-dimensional compact spaces

Abstract

We investigate properties of the clopen type semigroup of an action of a countable group on a compact, $0$-dimensional, Hausdorff space $X$. We discuss some characterizations of dynamical comparison (most of which were already known in the metrizable case) in this setting and prove that for a Cantor minimal action $\alpha$ of an amenable group the topological full group of $\alpha$ admits a dense, locally finite subgroup iff the corresponding clopen type semigroup is unperforated. We also discuss some properties of clopen type semigroups of the Stone–Čech compactifications and universal minimal flows of countable groups, and derive some consequences on generic properties in the space of minimal actions of a given countable group on the Cantor space.