Topology and convexity in the space of actions modulo weak equivalence

We analyse the structure of the quotient ${\mathrm{A}}_{\sim }(\mathrm{\Gamma },X,\mu )$ 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 ${\mathrm{A}}_{\sim }(\mathrm{\Gamma },X,\mu )$ is compatible with the topology, and as a consequence deduce that ${\mathrm{A}}_{\sim }(\mathrm{\Gamma },X,\mu )$ is path connected. Using ideas of Tucker-Drob we are able to give a complete description of the topological and convex structure of ${\mathrm{A}}_{\sim }(\mathrm{\Gamma },X,\mu )$ for amenable $\mathrm{\Gamma }$ by identifying it with the simplex of invariant random subgroups. In particular we conclude that ${\mathrm{A}}_{\sim }(\mathrm{\Gamma },X,\mu )$ can be represented as a compact convex subset of a Banach space if and only if $\mathrm{\Gamma }$ is amenable. We consider the space ${\mathrm{A}}_{{\sim }_s}(\mathrm{\Gamma },X,\mu )$ 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{F}\mathrm{R}$_{\sim_s}($\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 ${\mathrm{F}\mathrm{R}}_{{\sim }_s}({\mathbb{F}}_N,X,\mu )$.