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Coamenability and cospectral radius for orbit equivalence relations

Ben Hayes

Abstract

We consider inclusions $\mathcal{S}\leq \mathcal{R}$ 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 $\mathcal{S}\leq \mathcal{R}$. We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.

Coamenability and cospectral radius for orbit equivalence relations

Abstract

We consider inclusions 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 -classes. In this paper, we investigate the connections of this cospectral radius to the coamenability of the inclusion . We also undertake a systematic study of coamenability for inclusions of relations, establishing several equivalence formulations of this notion.

Paper Structure

This paper contains 19 sections, 39 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Remark
  3. Organization of the paper
  4. Preliminaries
  5. Background definitions and notational conventions
  6. Basic construction for equivalence relations
  7. Background on cospectral radius
  8. Generalities on invariant means, Fø lner sets, and almost invariant vectors
  9. coamenability for relations
  10. coamenability and almost sure cospectral radius
  11. Simplification of the cospectral radius condition under spectral gap
  12. Connections to percolation theory
  13. Coamenability in terms of von Neumann algebras
  14. Preliminaries on von Neumann algebras
  15. Preliminaries on direct integrals
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let ${\mathcal{S}}\leq {\mathcal{R}}$ be discrete, probability measure-preserving equivalence relations on $(X,\mu)$ with ${\mathcal{R}}$ ergodic. Then the following are equivalent:

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Corollary 2.3
  • Definition 2.4
  • Theorem 2.5
  • ...and 67 more