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A differential complex for CAT(0) cubical spaces

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.
0001-8708
1054-1111
Brodzki, Jacek
b1fe25fd-5451-4fd0-b24b-c59b75710543
Guentner, Erik
0efa2b74-da7d-497d-8a80-e668eb8f41f1
Higson, Nigel
fdac8f8c-825f-482c-9ea1-b97e956a2b24
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, 1054-1111. (doi:10.1016/j.aim.2019.03.009).

Record type: Article

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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Accepted/In Press date: 27 February 2019
e-pub ahead of print date: 14 March 2019
Published date: 30 April 2019
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 401823
URI: https://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: 14 Aug 2019 18:05

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