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

A cohomological characterisation of Yu's Property A for metric spaces
A cohomological characterisation of Yu's Property A for metric spaces
We introduce the notion of an asymptotically invariant mean as a coarse averaging operator for a metric space and show that the existence of such an operator is equivalent to Yu’s property A. As an application we obtain a positive answer to Higson’s question concerning the existence of a cohomological characterisation of property A. Specifically we provide coarse analogues of group cohomology and bounded cohomology (controlled cohomology and asymptotically invariant cohomology, respectively) for a metric space X, and provide a cohomological characterisation of property A which generalises the results of Johnson and Ringrose describing amenability in terms of bounded cohomology. These results amplify Guentner’s observation that property A should be viewed as coarse amenability for a metric space. We further provide a generalisation of Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This is used to derive Nowak’s theorem that the union of finite cubes of all dimensions does not have property A.
amenability, property A, bounded cohomology, invariant means, box spaces
1465-3060
391-432
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd

Brodzki, Jacek, Niblo, Graham A. and Wright, Nick (2012) A cohomological characterisation of Yu's Property A for metric spaces. Geometry & Topology, 16 (1), 391-432. (doi:10.2140/gt.2012.16.391).

Record type: Article

Abstract

We introduce the notion of an asymptotically invariant mean as a coarse averaging operator for a metric space and show that the existence of such an operator is equivalent to Yu’s property A. As an application we obtain a positive answer to Higson’s question concerning the existence of a cohomological characterisation of property A. Specifically we provide coarse analogues of group cohomology and bounded cohomology (controlled cohomology and asymptotically invariant cohomology, respectively) for a metric space X, and provide a cohomological characterisation of property A which generalises the results of Johnson and Ringrose describing amenability in terms of bounded cohomology. These results amplify Guentner’s observation that property A should be viewed as coarse amenability for a metric space. We further provide a generalisation of Guentner’s result that box spaces of a finitely generated group have property A if and only if the group is amenable. This is used to derive Nowak’s theorem that the union of finite cubes of all dimensions does not have property A.

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Accepted/In Press date: 11 November 2011
Published date: 7 March 2012
Keywords: amenability, property A, bounded cohomology, invariant means, box spaces
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 159393
URI: https://eprints.soton.ac.uk/id/eprint/159393
ISSN: 1465-3060
PURE UUID: 587e5525-d181-4f54-a6b6-b2e419dc3e90
ORCID for Graham A. Niblo: ORCID iD orcid.org/0000-0003-0648-7027
ORCID for Nick Wright: ORCID iD orcid.org/0000-0003-4884-2576

Catalogue record

Date deposited: 29 Jun 2010 21:34
Last modified: 24 Sep 2019 00:57

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