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Mixing actions of countable groups are almost free

Author: Robin D. Tucker-Drob
Journal: Proc. Amer. Math. Soc. 143 (2015), 5227-5232
MSC (2010): Primary 37A15, 37A25; Secondary 20F50
Published electronically: August 20, 2015
MathSciNet review: 3411140
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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.

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Additional Information

Robin D. Tucker-Drob
Affiliation: Department of Mathematics, Rutgers University, Piscataway, New Jersey 08854

Keywords: Mixing, total ergodicity, essentially free action, non-free action, Bernoulli shift, Bernoulli factor
Received by editor(s): September 2, 2012
Received by editor(s) in revised form: February 4, 2014
Published electronically: August 20, 2015
Communicated by: Nimish Shah
Article copyright: © Copyright 2015 American Mathematical Society

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