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

TL;DR

The work identifies a sharp equivalence between group exactness and the amenability of weak limits of finite invariant random equivalence relations, giving a precise criterion: $\overline{\mathrm{FIRE}}_\Gamma = \mathrm{AIRE}_\Gamma$ if and only if $\Gamma$ is exact. The authors develop FIREs and AIREs, and introduce classwise coherent $X$-markings to transfer boundary amenability into limits, proving exact groups have all weak finite-IRE limits amenable. Conversely, for nonexact groups, they construct weak limits of FIREs that are not amenable by embedding small scale expanders and exploiting their obstruction to hyperfiniteness, with a detailed mechanism via IREs from finite subsets and coinduction. The results connect to discrete analogues of ideal Poisson/Bernoulli Voronoi tessellations and yield a discrete boundary-amenability viewpoint on exactness, enriching the interplay between graph limits, amenability, and invariant random structures.

Abstract

We characterize exactness of a countable group $Γ$ in terms of invariant random equivalence relations (IREs) on $Γ$. Specifically, we show that $Γ$ 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 tesselation.

Theorems & Definitions (42)

• Theorem 1.1: see Theorem \ref{['thm:3']}
• Theorem 1.2: see Theorem \ref{['thm:exactWFIREisAIRE']}
• Corollary 1.3
• Definition 2.1
• Definition 2.2
• Theorem 2.3: Brodzki--Niblo--Špakula--Willett--Wright ULA, Elek ElekUla
• Definition 2.4
• Lemma 2.5
• proof
• Example 2.6: Cayley graphs