Mixing actions of countable groups are almost free

TL;DR

This paper demonstrates that totally ergodic actions of countable groups have stabilizers equal to a finite normal subgroup, linking ergodic theory with group structure, and characterizes groups with free Bernoulli factors.

Contribution

It establishes a connection between totally ergodic actions and finite normal subgroups, providing a new group-theoretic characterization of groups with free Bernoulli factors.

Findings

Existence of a finite normal subgroup for totally ergodic actions.

Characterization of groups with all non-trivial Bernoulli factors being free.

Use of group-theoretic results involving infinite locally finite groups.

Abstract

A measure preserving action of a countably infinite group \Gamma is called totally ergodic if every infinite subgroup of \Gamma acts ergodically. For example, all mixing and mildly mixing actions are totally ergodic. This note shows that if an action of \Gamma is totally ergodic then there exists a finite normal subgroup N of \Gamma such that the stabilizer of almost every point is equal to N. Surprisingly the proof relies on the group theoretic fact (proved by Hall and Kulatilaka as well as by Kargapolov) that every infinite locally finite group contains an infinite abelian subgroup, of which all known proofs rely on the Feit-Thompson theorem. As a consequence we deduce a group theoretic characterization of countable groups whose non-trivial Bernoulli factors are all free: these are precisely the groups that posses no finite normal subgroup other than the trivial subgroup.