A Differential Complex for CAT(0) Cubical Spaces

TL;DR

This paper extends a Fredholm operator construction from trees to CAT(0) cubical spaces, enabling new applications in operator K-theory and group representation theory.

Contribution

It generalizes existing operators to higher-dimensional CAT(0) cubical spaces, providing tools for analyzing group actions and K-amenability.

Findings

Extended operator construction to CAT(0) cubical spaces

Provided a new proof of K-amenability for certain groups

Demonstrated applications in operator K-theory

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.