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

TL;DR

This paper characterizes the exactness of countable groups through the properties of invariant random equivalence relations, linking group exactness to amenability in the space of IREs.

Contribution

It provides a new characterization of group exactness using invariant random equivalence relations and their limits, connecting to amenability and geometric tessellations.

Findings

A countable group is exact iff every weak limit of finite IREs is amenable.

For exact groups, the restricted rerooting relation is amenable.

The results relate group properties to geometric tessellations like Voronoi diagrams.

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.