Weak equivalence and non-classifiability of measure preserving actions

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.