MIXING ACTIONS OF COUNTABLE GROUPS ARE ALMOST FREE

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.