Journal of Noncommutative Geometry


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Volume 8, Issue 1, 2014, pp. 45–59
DOI: 10.4171/JNCG/148

Published online: 2014-03-20

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

Ronghui Ji[1], Crichton Ogle[2] and Bobby W. Ramsey[3]

(1) Indiana University-Purdue University, Indianapolis, USA
(2) Ohio State University, Columbus, USA
(3) University of Hawai‘i at Mānoa, USA

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 $\mathrm{HH}_*^t({\mathcal{H}_{\mathcal{B},L}(G)})_{\langle x\rangle}$ and $\mathrm{HC}_*^t({\mathcal{H}_{\mathcal{B},L}(G)})_{\langle x\rangle}$ for any bounding class $\mathcal{B}$, discrete group with word-length $(G,L)$ and conjugacy class $\langle x\rangle\in \langle G\rangle$. 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.

Keywords: $\mathcal{B}$-bounded cohomology, isocohomological, weighted complex, Generalized Bass Conjecture

Ji Ronghui, Ogle Crichton, Ramsey Bobby: On the Hochschild and cyclic (co)homology of rapid decay group algebras. J. Noncommut. Geom. 8 (2014), 45-59. doi: 10.4171/JNCG/148