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arXiv · 1610.05069

A Differential Complex for CAT(0) Cubical Spaces

Jacek Brodzki·Erik Guentner·Nigel Higson
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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 coccyges. 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 bounded-geometry CAT(0)-cubical spaces.

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Jacek Brodzki, Erik Guentner, Nigel Higson. 2016-10-17. A Differential Complex for CAT(0) Cubical Spaces. https://arxiv.org/abs/1610.05069

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