On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866912238608056320 |
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| author | Brück, Benjamin Miller, Jeremy Patzt, Peter Sroka, Robin J. Wilson, Jennifer C. H. |
| author_facet | Brück, Benjamin Miller, Jeremy Patzt, Peter Sroka, Robin J. Wilson, Jennifer C. H. |
| contents | Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to $\mathrm{SL}_n(\mathbb{Q})$. We determine the "relations among the relations" of this Steinberg module. That is, we construct an explicit partial resolution of length two of the $\mathrm{SL}_n(\mathbb{Z})$-module $\mathrm{St}_n(\mathbb{Q})$. We use this partial resolution to show the codimension-2 rational cohomology group $H^{{n \choose 2} -2}(\mathrm{SL}_n(\mathbb{Z});\mathbb{Q})$ of $\mathrm{SL}_n(\mathbb{Z})$ vanishes for $n \geq 3$. This resolves a case of a conjecture of Church-Farb-Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_11967 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$ Brück, Benjamin Miller, Jeremy Patzt, Peter Sroka, Robin J. Wilson, Jennifer C. H. Algebraic Topology Group Theory Geometric Topology Number Theory 11F75, 55U10, 55-08 Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual duality group of dimension $n \choose 2$ and the Steinberg module $\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to $\mathrm{SL}_n(\mathbb{Q})$. We determine the "relations among the relations" of this Steinberg module. That is, we construct an explicit partial resolution of length two of the $\mathrm{SL}_n(\mathbb{Z})$-module $\mathrm{St}_n(\mathbb{Q})$. We use this partial resolution to show the codimension-2 rational cohomology group $H^{{n \choose 2} -2}(\mathrm{SL}_n(\mathbb{Z});\mathbb{Q})$ of $\mathrm{SL}_n(\mathbb{Z})$ vanishes for $n \geq 3$. This resolves a case of a conjecture of Church-Farb-Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$. |
| title | On the codimension-two cohomology of $\mathrm{SL}_n(\mathbb{Z})$ |
| topic | Algebraic Topology Group Theory Geometric Topology Number Theory 11F75, 55U10, 55-08 |
| url | https://arxiv.org/abs/2204.11967 |