The space of stable weak equivalence classes of measure-preserving actions

The concept of (stable) weak containment for measure preserving actions of a countable group Gamma is analogous to the classical notion of (stable) weak containment of unitary representations. If Gamma is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if Gamma is non-amenable then there can be many different weak and stable weak equivalence classes. Our main result is that the set of stable weak equivalence classes naturally admits the structure of a Choquet simplex. For example, when Gamma = Z this simplex has only a countable set of extreme points but when Gamma is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when Gamma contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points. (C) 2017 Elsevier Inc. All rights reserved.