Uniform Local Amenability implies Property A

Elek, Gabor (2021) Uniform Local Amenability implies Property A. Proceedings of the American Mathematical Society, 149. pp. 2573-2577. ISSN 0002-9939

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Abstract

In this short note we answer a query of Brodzki, Niblo, \v{S}pakula, Willett and Wright \cite{ULA} by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser \cite{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 $\cap H_i=\{e_\Gamma\}$, and the associated Schreier graph sequence is of Property A.