Uniform local amenability implies Property A

Abstract

In this short note we answer a query of Brodzki, Niblo, Spakula, Willett and Wright [2] by showing that all bounded degree uniformly locally amenable graphs have Property A. For the second result of the note recall that Kaiser [9] proved that if Γ is a finitely generated group and {H i } ∞ 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 } ∞ i=1 such that ∩H i = {e Γ }, and the associated Schreier graph sequence is of Property A.

Planarity can be verified by an approximate proof labeling scheme in constant-time

“…The strong hyperfiniteness of monotone hyperfinite classes plays an important role in the proof of Theorem 1. Note that the author later proved in [4] that the notions of Property A, uniform hyperfiniteness and strong hyperfiniteness in fact coincide.…”

“…→ Prob(G) be a probability measure valued function witnessing the fact that G ∈ A d ε,r , that is, • for any adjacent pair of vertices x, y ∈ V (G) (4) g(x) − g(y) 1 ≤ ε .…”

Expanders are counterexamples to the $\ell^p$ coarse Baum–Connes conjecture

“…Combining Proposition 3.4 with [53, Theorem 5.1], we know that a bounded geometry metric space with the 2-operator norm localization property will have the p-operator norm localization property for all p 2 OE1; 1/. We do not know whether the converse holds, but we thank the referee for suggesting a possible approach to this problem, namely by considering uniform local amenability along the lines of [5,15].…”

Uniform hyperfiniteness

“…One can prove [4] that strong hyperfiniteness is also equivalent to Property A. The main idea of the proof of Theorem 2 is to show that µ-uniform hyperfiniteness is equivalent to a measured version of weighted hyperfiniteness, which, in turn, is equivalent to some measured versions of strong hyperfiniteness and finally, they all are equivalent to µ-uniform amenability.…”

“…Proof. We closely follow the combinatorial proof of Lemma 4.1 in [4]. Let C be the set of elements y ∈ L 2 (X, µ) which can be written in the form…”