Mixing actions of countable groups are almost free

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 possess no finite normal subgroup other than the trivial subgroup.