Equivariant $K$-theory and the Chern character for discrete groups

Abstract:

Let $X$ be a compact Hausdorff space, let $\Gamma$ be a discrete group that acts continuously on $X$ from the right, define $\widetilde {X} = \{(x,\gamma ) \in X \times \Gamma : x\cdot \gamma = x\}$, and let $\Gamma$ act on $\widetilde {X}$ via the formula $(x,\gamma )\cdot \alpha = (x\cdot \alpha , \alpha ^{-1}\gamma \alpha )$. Results of P. Baum and A. Connes, along with facts about the Chern character, imply that $K^i_\Gamma (X)$ and $K^i(\widetilde {X}/ \Gamma )$ are isomorphic up to torsion for $i = 0, 1$. In this paper, we present an example where the groups $K^i_\Gamma (X)$ and $K^i(\widetilde {X}/ \Gamma )$ are not isomorphic.