Hyper-u-amenablity and Hyperfiniteness of Treeable Equivalence Relations

TL;DR

This paper introduces new notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, demonstrating their implications for hyperfiniteness in treeable cases and establishing conditions under which such relations are hyperfinite.

Contribution

It defines strong forms of amenability for Borel equivalence relations and proves that treeable, hyper-u-amenable relations are hyperfinite, extending understanding of their structure.

Findings

Treeable, hyper-u-amenable relations are hyperfinite.

Measure-hyperfinite relations from free group actions are hyperfinite.

Certain amenable and bounded treeable relations are hyperfinite.

Abstract

We introduce the notions of u-amenability and hyper-u-amenability for countable Borel equivalence relations, strong forms of amenability that are implied by hyperfiniteness. We show that treeable, hyper-u-amenable countable Borel equivalence relations are hyperfinite. One of the corollaries that we get is that if a countable Borel equivalence relation is measure-hyperfinite and equal to the orbit equivalence relation of a free continuous action of a virtually free group on a $\sigma$-compact Polish space, then it is hyperfinite. We also obtain that if a countable Borel equivalence relation is treeable and equal to the orbit equivalence relation of a Borel action of an amenable group on a standard Borel space, or if it is treeable, amenable and Borel bounded, then it is hyperfinite.