Exactness and the topology of the space of invariant random equivalence relations

Abstract

We characterize exactness of a countable group $\Gamma$ in terms of invariant random equivalence relations (IREs) on $\Gamma$. Specifically, we show that $\Gamma$ is exact if and only if every weak limit of finite IREs is an amenable IRE. In particular, for exact groups this implies amenability of the restricted rerooting relation associated to the ideal Bernoulli Voronoi tessellation, the discrete analog of the ideal Poisson Voronoi tessellation.