K-homological finiteness and hyperbolic groups

Abstract

Motivated by classical facts concerning closed manifolds, we introduce a strong finiteness property in K-homology. We say that a C*-algebra has uniformly summable K-homology if all its K-homology classes can be represented by Fredholm modules which are finitely summable over the same dense subalgebra, and with the same degree of summability. We show that two types of C*-algebras associated to hyperbolic groups - the C*-crossed product for the boundary action, and the reduced group C*-algebra - have uniformly summable K-homology. We provide explicit summability degrees, as well as explicit finitely summable representatives for the K-homology classes.