Uniform hyperfiniteness

TL;DR

This paper introduces the concept of uniform hyperfiniteness for measurable graph equivalence relations, proving its equivalence to other hyperfiniteness notions and linking it to group exactness.

Contribution

It defines uniform hyperfiniteness for bounded degree graphs and shows its equivalence to weighted and fractional hyperfiniteness, extending classical results.

Findings

Uniform hyperfiniteness coincides with weighted and fractional hyperfiniteness.

Characterizes exactness of finitely generated groups via uniform hyperfiniteness.

Establishes a uniform version of hyperfiniteness for measurable graph relations.

Abstract

Almost forty years ago, Connes, Feldman and Weiss proved that for measurable equivalence relations the notions of amenability and hyperfiniteness coincide. In this paper we define the uniform version of amenability and hyperfiniteness for measurable graphed equivalence relations of bounded vertex degrees and prove that these two notions coincide as well. Roughly speaking, a measured graph $\cG$ is uniformly hyperfinite if for any $\eps>0$ there exists $K\geq 1$ such that not only $\cG$, but all of its subgraphs of positive measure are $(\eps,K)$-hyperfinite. We also show that this condition is equivalent to weighted hyperfiniteness and a strong version of fractional hyperfiniteness, a notion recently introduced by Lov\'asz. As a corollary, we obtain a characterization of exactness of finitely generated groups via uniform hyperfiniteness.