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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
A cohomological characterisation of Yu's property A for metric spaces

Jacek Brodzki, Graham A Niblo and Nick Wright

Geometry & Topology 16 (2012) 391–432
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 –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.

Keywords
Property A, bounded cohomology, coarse geometry, group cohomology
Mathematical Subject Classification 2000
Primary: 55N91
Secondary: 20J06, 30L05
References
Publication
Received: 27 May 2011
Accepted: 11 November 2011
Published: 7 March 2012
Proposed: Danny Calegari
Seconded: Peter Teichner, Steve Ferry
Authors
Jacek Brodzki
School of Mathematics
University of Southampton
Highfield
Southampton
SO17 1SH
England
Graham A Niblo
School of Mathematics
University of Southampton
Highfield
Southampton
SO17 1SH
England
Nick Wright
School of Mathematics
University of Southampton
Highfield
Southampton
SO17 1SH
England