Relatively hyperbolic groups, rapid decay algebras, and a generalization of the Bass conjecture

Abstract

By deploying dense subalgebras of ℓ1(G) we generalize the Bass conjecture in terms of Connes’ cyclic homology theory. In particular, we propose a stronger version of the ℓ1-Bass Conjecture. We prove that hyperbolic groups relative to finitely many subgroups, each of which posses the polynomial conjugacy bound property and nilpotent periodicity property, satisfy the ℓ1-Stronger-Bass Conjecture. Moreover, we determine the conjugacy bound for relatively hyperbolic groups and compute the cyclic cohomology of the ℓ1-algebra of any discrete group.