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Character theory and Euler characteristic for orbispaces and infinite groups

Wolfgang Lück, Irakli Patchkoria, Stefan Schwede

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 $Γ_\frac{p-1}{2}$ 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 $K$.

Character theory and Euler characteristic for orbispaces and infinite groups

Abstract

Given a discrete group with a finite model for , we study and , where is the -th Morava -theory for a given prime and is the height Morava -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 and the -theoretic Euler characteristic of in terms of centralizers. In certain cases these calculations lead to a full computation of , for example when is a right angled Coxeter group, and for . We apply our results to the mapping class group for an odd prime and to certain arithmetic groups, including the symplectic group for an odd prime and for a totally real field .

Paper Structure

This paper contains 13 sections, 29 theorems, 204 equations.

Table of Contents

  1. Orbispaces
  2. Orbispaces and Morava K-theory
  3. Hopkins--Kuhn--Ravenel character theory for orbispaces
  4. Connections to orbifold Euler characteristic
  5. Computations and examples
  6. Right angled Coxeter groups
  7. The special linear group SL(3,Z
  8. The special linear group SL(2,O for a totally real field K
  9. Crystallographic groups
  10. The general linear group GL(p-1,Z) for a prime p>=5
  11. The special linear group SL(p-1,Z) for a prime p>=5
  12. The symplectic group Sp(p-1,Z) for a prime p>=5
  13. The mapping class group Gamma for a prime p>=5

Key Result

Proposition 1.6

Let $G$ be a discrete group. Let $f:A\longrightarrow B$ be a continuous $G$-map between $G$-spaces such that for every finite subgroup $H$ of $G$, the map of fixed points $f^H:A^H\longrightarrow B^H$ is a weak equivalence. Then the morphism $G\backslash\!\!\backslash f:G\backslash\!\!\backslash A\lo

Theorems & Definitions (85)

  • Definition 1.4
  • Example 1.5: Global quotient orbispaces
  • Proposition 1.6
  • proof
  • Example 1.7: Global classifying spaces
  • Example 1.8
  • Example 1.9
  • Definition 1.10
  • Example 1.11
  • Theorem 1.12
  • ...and 75 more