Positive sofic entropy implies finite stabilizer

TL;DR

This paper demonstrates that measure-preserving actions of sofic groups with positive sofic entropy have a significant subset of points with finite stabilizers, extending known results to a broader class of groups.

Contribution

It extends the understanding of stabilizer properties in sofic group actions and links positive sofic entropy to the freeness of the action.

Findings

Positive sofic entropy implies positive measure of points with finite stabilizer

Inner automorphism actions of sofic groups have zero topological sofic entropy

Faithful actions with completely positive sofic entropy are free

Abstract

We prove that for a measure preserving action of a sofic group with positive sofic entropy, the set of points with finite stabilizer have positive measure. This extends results of Weiss and Seward for amenable groups and free groups, respectively. It follows that the action of a sofic group on its subgroups by inner automorphisms has zero topological sofic entropy, and a faithful action with completely positive sofic entropy must be free.