Hochschild homology relative to a family of groups

Abstract

We define the Hochschild homology groups of a group ring $\mathbb{Z}G$ relative to a family of subgroups $\mathcal{ℱ}$ of $G$. These groups are the homology groups of a space which can be described as a homotopy colimit, or as a configuration space, or, in the case $\mathcal{ℱ}$ is the family of finite subgroups of $G$, as a space constructed from stratum preserving paths. An explicit calculation is made in the case $G$ is the infinite dihedral group.