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

## TL;DR

This paper studies the structure of stable weak equivalence classes of measure-preserving actions of countable groups, revealing a rich geometric structure that varies with the group's properties, including the presence of free subgroups.

## Contribution

It proves that the set of stable weak equivalence classes forms a Choquet simplex, with its structure depending on whether the group is amenable or contains free subgroups.

## Key findings

- For amenable groups, all essentially free actions are weakly equivalent.
- For non-amenable free groups, the simplex is the Poulsen simplex.
- Groups containing nonabelian free groups have uncountably many strongly ergodic extreme points.

## Abstract

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 $\Gamma=\mathbb{Z}$ 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.