The cyclic homology of the group rings (Q1077511)

Let G be a discrete group. For \(x\in G\), denote by \(G_ x\) the centralizer of x and by \(N_ x\) the quotient of \(G_ x\) by the subgroup generated by x. Write \(<G>=<G>_ f\cup <G>_{\infty}\) for the decomposition of the conjugacy classes of G into classes \(\bar x\) with the property that ord x\(<\infty\), resp. ord x\(=\infty\). Main result: \[ 1)\quad HH_*(k[G])=\oplus_{\bar x\in <G>}H_*(BG_ x;k), \] \[ 2)\quad HC_*(k[G])=\oplus_{\bar x\in <G>_ f}H_*(BN_ x;k)\otimes H_*(BS^ 1;k)+\oplus_{x\bar {\;}\in <G>_{\infty}}H_*(BN_ x;k). \] Here k is a field of characteristic zero and \(HH_*\) resp. \(HC_*\) denote the Hochschild resp. cyclic homology groups, see \textit{A. Connes} [C. R. Acad. Sci., Paris, Sér. I 296, 953-958 (1983; Zbl 0534.18009)]. This nice computation, which indicates the different types of contribution to \(HC_*\) of classes of finite resp. infinite order, is carried out by identifying the whole Connes-Gysin sequence of k[G] with the one obtained starting from a certain cyclic set associated with G and then noticing a splitting (as a cyclic set) of this construction, which is parametrized by \(<G>\). The technical device is the notion of a cyclic groupoid. Along the way, similar results are derived for \(k=arbitrary\) commutative ring. As an application, Künneth-type formulae are offered for \(HC_*(k[G*H])\) and \(HC_*(k[G\times {\mathbb{Z}}])\).