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

TL;DR

This paper characterizes the amenability of countable Borel equivalence relations via the uniform Liouville property, explores Kesten's property for topological groups, and constructs an example of an amenable group lacking Kesten's property.

Contribution

It provides a new characterization of amenability for Borel equivalence relations and links Kesten's property to random walk behavior on these structures.

Findings

Amenability characterized by the uniform Liouville property.

Amenable topological groups with small invariant neighborhoods have Kesten's property.

Existence of an amenable contractible Polish group without Kesten's property.

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.