Uniform Local Amenability implies Property A

Abstract

In this short note we answer a query of Brodzki, Niblo, Špakula, 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 $\{H_i{\}}_{i=1}^{\infty }$ 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 $\{H_i{\}}_{i=1}^{\infty }$ such that $\cap H=\{e_{\Gamma }\}$, and the associated Schreier graph sequence is of Property A.

to appear in the Proceedings of the American Mathematical Society