UNIFORM LOCAL AMENABILITY IMPLIES PROPERTY A

In this short note we answer a query of Brodzki, Niblo, Spakula, 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 F is a finitely generated group and {H-i}(i=1)(infinity) 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)(infinity) such that boolean AND H-i = {e(Gamma)}, and the associated Schreier graph sequence is of Property A.