Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909765350719488 |
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| author | Miller, Jeremy Patzt, Peter Putman, Andrew |
| author_facet | Miller, Jeremy Patzt, Peter Putman, Andrew |
| contents | We prove that the homology groups of any connected reductive group over a field with coefficients in the Steinberg representation vanish in a range. The generalizes work of Ash-Putman-Sam on the classical split groups. We state a connectivity conjecture that would allow us to prove such a vanishing result for $SL_n(\mathbb{Z})$, as was conjectured by Church-Farb-Putman. We prove some special cases of this conjecture and use it to refine known results about the first and second of homology of $SL_n(\mathbb{Z})$ with Steinberg coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_01559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures Miller, Jeremy Patzt, Peter Putman, Andrew Algebraic Topology Group Theory Representation Theory We prove that the homology groups of any connected reductive group over a field with coefficients in the Steinberg representation vanish in a range. The generalizes work of Ash-Putman-Sam on the classical split groups. We state a connectivity conjecture that would allow us to prove such a vanishing result for $SL_n(\mathbb{Z})$, as was conjectured by Church-Farb-Putman. We prove some special cases of this conjecture and use it to refine known results about the first and second of homology of $SL_n(\mathbb{Z})$ with Steinberg coefficients. |
| title | Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures |
| topic | Algebraic Topology Group Theory Representation Theory |
| url | https://arxiv.org/abs/2509.01559 |