Character theory and Euler characteristic for orbispaces and infinite groups

Abstract

Given a discrete group $G$ with a finite model for $\underline{E}G$, we study $K(n{)}^{\ast }(BG)$ and $E^{\ast }(BG)$, where $K(n)$ is the $n$-th Morava $K$-theory for a given prime and $E$ is the height $n$ Morava $E$-theory. In particular we generalize the character theory of Hopkins, Kuhn and Ravenel who studied these objects for finite groups. We give a formula for a localization of $E^{\ast }(BG)$ and the $K(n)$-theoretic Euler characteristic of $BG$ in terms of centralizers. In certain cases these calculations lead to a full computation of $E^{\ast }(BG)$, for example when $G$ is a right angled Coxeter group, and for $G=S{L}_3(\mathbb{Z})$. We apply our results to the mapping class group $\hat{I}{“}_{\frac{p-1}{2}}$ for an odd prime $p$ and to certain arithmetic groups, including the symplectic group $S{p}_{p-1}(\mathbb{Z})$ for an odd prime $p$ and $S{L}_2({\mathcal{O}}_K)$ for a totally real field $K$.

50 pages