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
391-432
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Niblo, Graham A.
43fe9561-c483-4cdf-bee5-0de388b78944
Wright, Nick
f4685b8d-7496-47dc-95f0-aba3f70fbccd
7 March 2012
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), .
(doi:10.2140/gt.2012.16.391).
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.
Text
The_meaning_of_Yu_s_property_A_Submission_Candidate_2.pdf
- Other
More information
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: http://eprints.soton.ac.uk/id/eprint/159393
ISSN: 1465-3060
PURE UUID: 587e5525-d181-4f54-a6b6-b2e419dc3e90
Catalogue record
Date deposited: 29 Jun 2010 21:34
Last modified: 14 Mar 2024 02:50
Export record
Altmetrics
Download statistics
Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.
View more statistics
