TY - JOUR

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

AU - Brodzki, J.

AU - Guentner, E.

AU - Higson, N.

N1 - Funding Information:
J. B. was supported in part by EPSRC grants EP/I016945/1 and EP/N014189/1.E. G. was supported in part by a grant from the Simons Foundation (#245398).N. H. was supported in part by NSF grant DMS-1101382.Acknowledgment. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme ‘Non-positive curvature, group actions and cohomology’ when part of the work on this paper was undertaken, supported by EPSRC Grant EP/K032208/1. The second author gratefully acknowledges support from the Simons Foundation during the programme.

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.

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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 -