Coamenability and cospectral radius for orbit equivalence relations

Abstract

We consider inclusions $\mathcal{S}\le \mathcal{R}$ of discrete, probability measure-preserving orbit equivalence relations. In previous work with Abért-Fraçzyk, 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 $\mathcal{S}\le \mathcal{R}$. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.

57 pages. No figures. Numerous edits, updated discussion on open problem on coamenability versus cohyperfiniteness, and generalized Proposition 7.2. Comments welcome!