A cohomological characterisation of Yu's property A for metric spaces

Abstract

We develop a new framework for cohomology of discrete metric spaces and groups which simultaneously generalises group cohomology, Roe’s coarse cohomology, Gersten’s ${\ell }^{\infty }$–cohomology and Johnson’s bounded cohomology. In this framework we give an answer to Higson’s question concerning the existence of a cohomological characterisation of Yu’s property A, analogous to Johnson’s characterisation of amenability. In particular, we introduce an analogue of invariant mean for metric spaces with property A. As an application we extend Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This provides an alternative proof of Nowak’s result that the infinite dimensional cube does not have property A.