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Amenable equivalence relations, Kesten's property, and measurable lamplighters

Maksym Chaudkhari, Kate Juschenko, Friedrich Martin Schneider

Abstract

We prove a characterization of the amenability of countable Borel equivalence relations in terms of the uniform Liouville property for group actions on their classes. Furthermore, inspired by a well-known amenability criterion for locally compact groups due to Kesten, we study return probabilities for random walks, and in particular a limiting condition that we call Kesten's property, on general topological groups. We show that every amenable topological group with small invariant neighborhoods indeed has Kesten's property. For measurable lamplighter groups associated with countable Borel equivalence relations, we establish a connection between Kesten's property and anti-concentration inequalities for the inverted orbits of random walks on the equivalence classes. This allows us to construct an amenable contractible Polish group without Kesten's property.

Amenable equivalence relations, Kesten's property, and measurable lamplighters

Abstract

We prove a characterization of the amenability of countable Borel equivalence relations in terms of the uniform Liouville property for group actions on their classes. Furthermore, inspired by a well-known amenability criterion for locally compact groups due to Kesten, we study return probabilities for random walks, and in particular a limiting condition that we call Kesten's property, on general topological groups. We show that every amenable topological group with small invariant neighborhoods indeed has Kesten's property. For measurable lamplighter groups associated with countable Borel equivalence relations, we establish a connection between Kesten's property and anti-concentration inequalities for the inverted orbits of random walks on the equivalence classes. This allows us to construct an amenable contractible Polish group without Kesten's property.
Paper Structure (10 sections, 20 theorems, 92 equations)

This paper contains 10 sections, 20 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Organization
  3. Preliminaries
  4. Amenability
  5. Equivalence relations and their full groups
  6. Uniform Liouville property
  7. Kesten's theorem for topological groups
  8. Extensive amenability and inverted orbits
  9. Measurable lamplighter groups
  10. Appendix

Key Result

Theorem A

Let $X$ be a standard Borel space equipped with a non-atomic Borel probability measure $\mu$, let $R$ be a countable Borel equivalence relation on $X$ such that $\mu$ is $R$-quasi-invariant, and let $G$ be a countable dense subgroup of the full group $[R]$ equipped with the uniform topology. The fol

Theorems & Definitions (46)

  • Theorem A: Theorem \ref{['UnifLiouville']}
  • Theorem B: Theorem \ref{['SINKesten']}
  • Theorem C: Theorem \ref{['Top.Lamplighter']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • ...and 36 more