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

By deploying dense subalgebras of `.G/ we generalize the Bass conjecture in terms of Connes’ cyclic homology theory. In particular, we propose a stronger version of the `-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 `-Stronger-Bass Conjecture. Moreover, we determine the conjugacy bound for relatively hyperbolic groups and compute the cyclic cohomology of the `-algebra of any discrete group. Mathematics Subject Classification (2010). 46L80, 20F65, 16S34.