Weak equivalence and non-classifiability of measure preserving actions

TL;DR

This paper proves a strong form of weak equivalence for measure-preserving actions of groups, shows non-classifiability of certain actions, and resolves questions about properties of residually finite groups using advanced ergodic theory techniques.

Contribution

It establishes a strong version of weak equivalence for measure-preserving actions, generalizes results to non-free actions, and answers open questions about properties of residually finite groups.

Findings

Weak equivalence of actions is strengthened to show s imes b is weakly equivalent to b.

Isomorphism on weak equivalence classes of free actions is not classifiable by countable structures.

Residually finite groups with properties EMD and MD are shown to be equivalent.

Abstract

Ab\'ert-Weiss have shown that the Bernoulli shift s of a countably infinite group \Gamma is weakly contained in any free measure preserving action (mpa) b of \Gamma. We establish a strong version of this result, conjectured by Ioana, by showing that s \times b is weakly equivalent to b. This is generalized to non-free mpa's using random Bernoulli shifts. The result for free mpa's is used to show that isomorphism on the weak equivalence class of a free mpa does not admit classification by countable structures. This provides a negative answer to a question of Ab\'ert and Elek. We also answer a question of Kechris regarding two ergodic theoretic properties of residually finite groups. An infinite residually finite group \Gamma is said to have EMD if the action p of \Gamma on its profinite completion weakly contains all ergodic mpa's of \Gamma, and \Gamma is said to have property MD if i \times p weakly contains all mpa's of \Gamma, where i denotes the trivial action on a standard non-atomic probability space. Kechris asks if these two properties equivalent and we provide a positive answer by studying the relationship between convexity and weak containment.