Character theory and Euler characteristic for orbispaces and infinite groups
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| Format: | Preprint |
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2024
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| author | Lück, Wolfgang Patchkoria, Irakli Schwede, Stefan |
| author_facet | Lück, Wolfgang Patchkoria, Irakli Schwede, Stefan |
| contents | 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 $K$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14510 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Character theory and Euler characteristic for orbispaces and infinite groups Lück, Wolfgang Patchkoria, Irakli Schwede, Stefan Algebraic Topology Group Theory K-Theory and Homology Primary: 55P42, 55R35. Secondary: 55P91, 19L, 20F65 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 $K$. |
| title | Character theory and Euler characteristic for orbispaces and infinite groups |
| topic | Algebraic Topology Group Theory K-Theory and Homology Primary: 55P42, 55R35. Secondary: 55P91, 19L, 20F65 |
| url | https://arxiv.org/abs/2410.14510 |