On the Hochschild and cyclic (co)homology of rapid decay group algebras

Abstract

We show that the technical condition of solvable conjugacy bound, introduced in [JOR1], can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands ${\mathrm{HH}}_{\ast }^t({\mathcal{H}}_{\mathcal{B},L}(G){)}_{⟨x⟩}$ and ${\mathrm{HC}}_{\ast }^t({\mathcal{H}}_{\mathcal{B},L}(G){)}_{⟨x⟩}$ for any bounding class $\mathcal{B}$, discrete group with word-length $(G,L)$ and conjugacy class $⟨x⟩\in ⟨G⟩$. We use this description to prove the conjecture $\mathcal{B}$-SrBC of [JOR1] for a class of groups that goes well beyond the cases considered in that paper. In particular, we show that the conjecture ${\ell }^1$-SrBC (the Strong Bass Conjecture for the topological K-theory of ${\ell }^1(G)$) is true for all semihyperbolic groups which satisfy SrBC, a statement consistent with the rationalized Bost conjecture for such groups.