Measures and stabilizers of group Cantor actions

TL;DR

This paper investigates the structure of stabilizers in minimal equicontinuous group actions on Cantor sets, providing conditions for the existence of a subgroup with full measure stabilizer conjugacy, and explores implications for invariant random subgroups.

Contribution

It introduces the concept of locally non-degenerate actions and establishes conditions for stabilizer conjugacy in Cantor actions, extending understanding of invariant random subgroups.

Findings

Existence of a subgroup with full measure stabilizer conjugacy under certain conditions

Introduction of the locally non-degenerate action condition

Application to invariant random subgroups and almost one-to-one extensions

Abstract

We consider a minimal equicontinuous action of a finitely generated group $G$ on a Cantor set $X$ with invariant probability measure $\mu$, and stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup $H$ of $G$ such that the set of points in $X$ whose stabilizers are conjugate to $H$ has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions.