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.
Highlights
CAT(0) cubical spaces are geometric spaces that are assembled from cubes of various dimensions in much the same way that simplicial trees are assembled from edges
They play an important role in group theory that is more or less analogous to the roles that trees play in the theory of amalgamated free products and HNN extensions, and they have been especially prominent in recent work in group theory related to 3-manifold topology [Ago14]
The goal of this paper and its sequel is to provide a geometric proof of the BaumConnes conjecture for groups acting on CAT(0)-cubical spaces
Summary
CAT(0) cubical spaces are geometric spaces that are assembled from cubes of various dimensions in much the same way that simplicial trees are assembled from edges. The homotopy has an important interpretation in operator K-theory: it connects the γ-element to the identity in Kasparov’s representation ring [Kas[88], Section 2], and so leads to a new, geometric proof of the Baum-Connes conjecture (with coefficients) for locally compact groups acting cocompactly and isometrically on a finite-dimensional CAT(0)-cubical space. Let us first describe the final, base-vertex-independent complex in the homotopy, which is built using the theory of hyperplanes in CAT(0) cubical spaces [NR98]. Their construction essentially completes the proof that γ = 1 for groups acting properly and isometrically on CAT(0) cubical spaces. There are similar fomulas involving parallelism classes of (p, q)-cubes for all q > 0, and they lead us to expect that aspects of our constructions will be of interest and value elsewhere in the theory of CAT(0) cube complexes
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