A differential complex for CAT(0) cubical spaces
A differential complex for CAT(0) cubical spaces
In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to C⁎-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on finite dimensional CAT(0)-cubical spaces.
1054-1111
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Guentner, Erik
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Higson, Nigel
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30 April 2019
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Guentner, Erik
0efa2b74-da7d-497d-8a80-e668eb8f41f1
Higson, Nigel
fdac8f8c-825f-482c-9ea1-b97e956a2b24
Brodzki, Jacek, Guentner, Erik and Higson, Nigel
(2019)
A differential complex for CAT(0) cubical spaces.
Advances in Mathematics, 347, .
(doi:10.1016/j.aim.2019.03.009).
Abstract
In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik and Ryszard Szwarc, constructed and studied a certain Fredholm operator associated to a simplicial tree. The operator can be defined in at least two ways: from a combinatorial flow on the tree, similar to the flows in Forman's discrete Morse theory, or from the theory of unitary operator-valued cocycles. There are applications of the theory surrounding the operator to C⁎-algebra K-theory, to the theory of completely bounded representations of groups that act on trees, and to the Selberg principle in the representation theory of p-adic groups. The main aim of this paper is to extend the constructions of Julg and Valette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is to illustrate the utility of the extended construction by developing an application to operator K-theory and giving a new proof of K-amenability for groups that act properly on finite dimensional CAT(0)-cubical spaces.
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2016-10-13-DifferentialComplex.pdf
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Accepted/In Press date: 27 February 2019
e-pub ahead of print date: 14 March 2019
Published date: 30 April 2019
Organisations:
Pure Mathematics
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Local EPrints ID: 401823
URI: http://eprints.soton.ac.uk/id/eprint/401823
ISSN: 0001-8708
PURE UUID: 388d72e5-1f53-4e60-aff6-0aee78e96726
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Date deposited: 24 Oct 2016 07:54
Last modified: 15 Mar 2024 03:11
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Author:
Erik Guentner
Author:
Nigel Higson
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