### 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 language | English (US) |
---|---|

Pages (from-to) | 1054-1111 |

Number of pages | 58 |

Journal | Advances in Mathematics |

Volume | 347 |

DOIs | |

State | Published - Apr 30 2019 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*347*, 1054-1111. https://doi.org/10.1016/j.aim.2019.03.009

}

*Advances in Mathematics*, vol. 347, pp. 1054-1111. https://doi.org/10.1016/j.aim.2019.03.009

**A differential complex for CAT(0) cubical spaces.** / Brodzki, J.; Guentner, E.; Higson, Nigel.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A differential complex for CAT(0) cubical spaces

AU - Brodzki, J.

AU - Guentner, E.

AU - Higson, Nigel

PY - 2019/4/30

Y1 - 2019/4/30

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85062860758&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85062860758&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2019.03.009

DO - 10.1016/j.aim.2019.03.009

M3 - Article

AN - SCOPUS:85062860758

VL - 347

SP - 1054

EP - 1111

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -