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

TL;DR

This paper removes a technical condition from previous work on Hochschild and cyclic (co)homology of rapid decay group algebras, providing a Burghelea-type description and proving the $ ext{SrBC}$ conjecture for a broad class of groups.

Contribution

It generalizes previous results by removing the solvable conjugacy bound condition and proves the $ ext{SrBC}$ conjecture for all semihyperbolic groups satisfying $ ext{SrBC}$.

Findings

Removed the solvable conjugacy bound condition.

Provided a Burghelea-type description for summands of Hochschild and cyclic (co)homology.

Proved the $ ext{ell}^1$-SrBC conjecture for all semihyperbolic groups satisfying SrBC.

Abstract

We show that the technical condition of solvable conjugacy bound, introduced in \cite{JOR1}, can be removed without affecting the main results of that paper. The result is a Burghelea-type description of the summands $HH_*^t(\BG)_{<x>}$ and $HC_*^t(\BG)_{<x>}$ for any bounding class $\B$, discrete group with word-length $(G,L)$ and conjugacy class $<x>\in <G>$. We use this description to prove the conjecture $\B$-SrBC of \cite{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.