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

Abstract

The concept of (stable) weak containment for measure-preserving actions of a countable group $Γ$ is analogous to the classical notion of (stable) weak containment of unitary representations. If $Γ$ is amenable then the Rokhlin lemma shows that all essentially free actions are weakly equivalent. However if $Γ$ 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 $Γ=\mathbb{Z}$ this simplex has only a countable set of extreme points but when $Γ$ is a nonamenable free group, this simplex is the Poulsen simplex. We also show that when $Γ$ contains a nonabelian free group, this simplex has uncountably many strongly ergodic essentially free extreme points.