Uniform Local Amenability implies Property A

TL;DR

This paper proves that bounded degree uniformly locally amenable graphs have Property A and constructs examples of non-amenable groups with Schreier graphs of Property A, addressing open questions in geometric group theory.

Contribution

It establishes that uniform local amenability implies Property A and provides counterexamples related to Schreier graphs and group amenability.

Findings

Uniform local amenability implies Property A for bounded degree graphs.

Existence of non-amenable groups with Schreier graphs of Property A.

Counterexamples to the converse of Kaiser’s result.

Abstract

In this short note we answer a query of Brodzki, Niblo, \v{S}pakula, Willett and Wright by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser proved that if $\Gamma$ is a finitely generated group and $\{H_i\}^\infty_{i=1}$ is a Farber sequence of finite index subgroups, then the associated Schreier graph sequence is of Property A if and only if the group is amenable. We show however, that there exist a non-amenable group and a nested sequence of finite index subgroups $\{H_i\}^\infty_{i=1}$ such that $\cap H=\{e_\Gamma\}$, and the associated Schreier graph sequence is of Property A.