Coamenability and cospectral radius for orbit equivalence relations

TL;DR

This paper explores the relationship between cospectral radius and coamenability in orbit equivalence relations, providing new equivalence characterizations of coamenability for inclusions.

Contribution

It establishes connections between cospectral radius and coamenability, and systematically studies coamenability for inclusions of orbit equivalence relations with new equivalence formulations.

Findings

Established the pointwise almost sure existence of the cospectral radius.

Connected cospectral radius to coamenability of inclusions.

Provided several new equivalence characterizations of coamenability.

Abstract

We consider inclusions $\mathcal{S}\leq \mathcal{R}$ of discrete, probability measure-preserving orbit equivalence relations. In previous work with Ab\'{e}rt-Fra\c{c}zyk, we established the pointwise almost sure existence of the cospectral radius of a random walk on the $\mathcal{R}$-classes. In this paper, we investigate the connections of this cospectral radius to the coamenability of the inclusion $\mathcal{S}\leq \mathcal{R}$. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.