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In the 1980's Pierre Julg and Alain Valette, and also Tadeusz Pytlik andRyszard Szwarc, constructed and studied a certain Fredholm operator associatedto a simplicial tree. The operator can be defined in at least two ways: from acombinatorial flow on the tree, similar to the flows in Forman's discrete Morsetheory, or from the theory of unitary operator-valued coccyges. There areapplications of the theory surrounding the operator to C*-algebra K-theory, tothe 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 andValette, and Pytlik and Szwarc, to CAT(0) cubical spaces. A secondary aim is toillustrate the utility of the extended construction by developing anapplication to operator K-theory and giving a new proof of K-amenability forgroups that act properly on bounded-geometry CAT(0)-cubical spaces.

A Differential Complex for CAT(0) Cubical Spaces