Topology and convexity in the space of actions modulo weak equivalence

TL;DR

This paper studies the topological and convex structure of the space of measure-preserving actions of groups, revealing connections with invariant random subgroups and characterizing when this space is a compact convex subset of a Banach space.

Contribution

It provides a complete description of the topological and convex structure of the space of actions for amenable groups and explores properties of stable weak equivalence classes for free groups.

Findings

The space of actions is path connected.

For amenable groups, the space is a simplex of invariant random subgroups.

Extreme points are dense in the space of stable weak equivalence classes for free groups.

Abstract

We analyse the structure of the quotient $\mathrm{A}_\sim(\Gamma,X,\mu)$ of the space of measure-preserving actions of a countable discrete group by the relation of weak equivalence. This space carries a natural operation of convex combination. We show that the convex structure of $\mathrm{A}_\sim(\Gamma,X,\mu)$ is compatible with the topology, and as a consequence deduce that $\mathrm{A}_\sim(\Gamma,X,\mu)$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of $\mathrm{A}_\sim(\Gamma,X,\mu)$ for amenable $\Gamma$ by identifying it with the simplex of invariant random subgroups. In particular we conclude that $\mathrm{A}_\sim(\Gamma,X,\mu)$ can be represented as a compact convex subset of a Banach space if and only if $\Gamma$ is amenable. We consider the space $\mathrm{A}_{\sim_s}(\Gamma,X,\mu)$ of stable weak equivalence classes and show that is always a compact convex subset of a Banach space. For a free group $\mathbb{F}_N$, we show that if one restricts to the compact convex set $\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,\mu) \subseteq \mathrm{A}_{\sim_s}(\mathbb{F}_N,X,\mu)$ of the stable weak equivalence classes of free actions, the extreme points are dense in $\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,\mu)$.