Character theory and Euler characteristic for orbispaces and infinite groups

TL;DR

This paper extends character theory and computes Morava K-theory and E-theory for classifying spaces of groups with finite models, providing explicit formulas and applications to various algebraic and geometric groups.

Contribution

It generalizes the character theory of Hopkins, Kuhn, and Ravenel to infinite groups with finite models, offering new formulas for Morava theories and Euler characteristics.

Findings

Derived formulas for localized $E^*(BG)$ and $K(n)$-theoretic Euler characteristics.

Computed $E^*(BG)$ explicitly for right angled Coxeter groups and $SL_3(\\mathbb{Z})$.

Applied results to mapping class groups and arithmetic groups like $Sp_{p-1}(\mathbb{Z})$.

Abstract

Given a discrete group $G$ with a finite model for $\underline{E}G$, we study $K(n)^*(BG)$ and $E^*(BG)$, 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 $E^*(BG)$ and the $K(n)$-theoretic Euler characteristic of $BG$ in terms of centralizers. In certain cases these calculations lead to a full computation of $E^*(BG)$, for example when $G$ is a right angled Coxeter group, and for $G=SL_3(\mathbb{Z})$. We apply our results to the mapping class group $\Gamma_\frac{p-1}{2}$ for an odd prime $p$ and to certain arithmetic groups, including the symplectic group $Sp_{p-1}(\mathbb{Z})$ for an odd prime $p$ and $SL_2(\mathcal{O}_K)$ for a totally real field $K$.