Controlled coarse homology and isoperimetric inequalities
Controlled coarse homology and isoperimetric inequalities
We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize the existence of primitives of volume forms with prescribed growth and show that coarse homology classes obstruct weighted Poincaré inequalities.
443-462
Nowak, Piotr W.
5772d57b-3f7b-4a94-9ffe-b71c568ae20c
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
1 May 2010
Nowak, Piotr W.
5772d57b-3f7b-4a94-9ffe-b71c568ae20c
Spakula, Jan
c43164e4-36a7-4372-9ce2-9bfbba775d77
Nowak, Piotr W. and Spakula, Jan
(2010)
Controlled coarse homology and isoperimetric inequalities.
Journal of Topology, 3 (2), .
(doi:10.1112/jtopol/jtq013).
Abstract
We study a coarse homology theory with prescribed growth conditions. For a finitely generated group G with the word length metric this homology theory turns out to be related to amenability of G. We characterize vanishing of a certain fundamental class in our homology in terms of an isoperimetric inequality on G and show that on any group at most linear control is needed for this class to vanish. The latter is a homological version of the classical Burnside problem for infinite groups, with a positive solution. As applications we characterize the existence of primitives of volume forms with prescribed growth and show that coarse homology classes obstruct weighted Poincaré inequalities.
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Accepted/In Press date: 11 November 2009
Published date: 1 May 2010
Identifiers
Local EPrints ID: 420843
URI: http://eprints.soton.ac.uk/id/eprint/420843
ISSN: 1753-8416
PURE UUID: 19b75752-6dfe-478f-a7ad-f6ed718ed59b
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Date deposited: 16 May 2018 16:30
Last modified: 16 Mar 2024 04:16
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Author:
Piotr W. Nowak
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