Uniform local amenability implies Property A

Abstract:

In this short note we answer a query of Brodzki, Niblo, Špakula, Willett and Wright [J. Noncommut. Geom. 7 (2013), pp. 583–603] by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser [Trans. Amer. Math. Soc. 372 (2019), pp. 2855–2874] 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 $\cap H_i=\{e_\Gamma \}$, and the associated Schreier graph sequence is of Property A.