Hyperfiniteness for group actions on trees

TL;DR

This paper explores conditions under which group actions on trees produce hyperfinite orbit equivalence relations on the boundary, providing criteria and examples for hyperfiniteness and non-hyperfiniteness.

Contribution

It introduces natural conditions for hyperfiniteness of boundary actions in group actions on trees and provides examples illustrating these conditions.

Findings

Acylindrical actions satisfy the hyperfiniteness condition.

A weakening of conditions leads to measure hyperfiniteness.

Examples of non-hyperfinite boundary actions are documented.

Abstract

We identify natural conditions for a countable group acting on a countable tree which imply that the orbit equivalence relation of the induced action on the Gromov boundary is Borel hyperfinite. Examples of this condition include acylindrical actions. We also identify a natural weakening of the aforementioned conditions that implies measure hyperfinitenss of the boundary action. We then document examples of group actions on trees whose boundary action is not hyperfinite.