Elasticity of free type III actions of free groups

TL;DR

This paper demonstrates that certain non-amenable, measure-class-preserving, type III equivalence relations can be realized by free actions of non-abelian free groups, including infinitely generated ones, with specific properties on Schreier graphs.

Contribution

It establishes a new connection between type III equivalence relations and free group actions, extending known results to the non-amenable, measure-class-preserving, type III setting.

Findings

Type III equivalence relations are induced by free actions of non-abelian free groups.

No ends of the Schreier graph are vanishing in these actions.

Highlights differences between type III and measure-preserving cases.

Abstract

We prove that measure-class-preserving non-amenable treeable equivalence relations of type III, meaning not preserving any equivalent $\sigma$-finite measure, are induced by free actions of non-abelian free groups of any given number of generators, including infinitely generated free groups, with the additional property that no ends of the induced Schreier graph are vanishing. This is done using a characterization of type III due to Hopf for transformations and Dang-Ngoc-Nghiem in general. This highlights the difference between the type III setting and the measure-preserving setting.