  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.
