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

TL;DR

This paper studies the properties of clopen type semigroups arising from actions of countable groups on 0-dimensional compact spaces, linking algebraic properties to dynamical features like minimality and comparison.

Contribution

It characterizes dynamical comparison in this setting and connects the unperforation of the semigroup to the density of certain subgroups in the topological full group.

Findings

Characterization of dynamical comparison for 0-dimensional actions

Equivalence between unperforation of semigroup and dense, locally finite subgroups

Insights into properties of semigroups for Stone-ech compactifications and minimal flows

Abstract

We investigate some 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-\v{C}ech 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.