A differential complex for CAT(0) cubical spaces

J. Brodzki, E. Guentner, Nigel Higson

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)1054-1111
Number of pages58
JournalAdvances in Mathematics
Volume347
DOIs
StatePublished - Apr 30 2019

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CAT(0)
K-theory
Operator
Discrete Morse Theory
P-adic Groups
Fredholm Operator
Amenability
Unitary Operator
Cocycle
Representation Theory
C*-algebra

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Brodzki, J. ; Guentner, E. ; Higson, Nigel. / A differential complex for CAT(0) cubical spaces. In: Advances in Mathematics. 2019 ; Vol. 347. pp. 1054-1111.
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A differential complex for CAT(0) cubical spaces. / Brodzki, J.; Guentner, E.; Higson, Nigel.

In: Advances in Mathematics, Vol. 347, 30.04.2019, p. 1054-1111.

Research output: Contribution to journalArticle

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