N ov 2 00 7 RELATIVELY HYPERBOLIC GROUPS , RAPID DECAY ALGEBRAS AND A GENERALIZATION OF THE BASS CONJECTURE

By deploying dense subalgebras of ℓ 1 (G) we generalize the Bass conjecture in terms of Connes' cyclic homology theory. In particular, we propose a stronger version of the ℓ 1-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 ℓ 1-Stronger-Bass Conjecture. Moreover, we determine the conjugacy-bound for relatively hyperbolic groups and compute the cyclic cohomology of the ℓ 1-algebra of any discrete group.