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

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