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arXiv · 2509.01559

Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures

Jeremy Miller·Peter Patzt·Andrew Putman
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Abstract

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.

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Jeremy Miller, Peter Patzt, Andrew Putman. 2025-09-01. Homological vanishing for the Steinberg representation II: reductive groups and integral conjectures. https://arxiv.org/abs/2509.01559

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