Topology and convexity in the space of actions modulo weak equivalence

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 $\mathrm{FR}_{\sim_s}(\mathbb{F}_N,X,\mu)$.