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Stability in the high-dimensional cohomology of congruence subgroups

Part of: Homology and cohomology theories Other groups of matrices Rings and algebras arising under various constructions Discontinuous groups and automorphic forms Whitehead groups and $K_1$

Published online by Cambridge University Press:  24 March 2020

Jeremy Miller ,
Rohit Nagpal  and
Peter Patzt
Jeremy Miller
Affiliation:
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, IN 47904, USA email jeremykmiller@purdue.edu
Rohit Nagpal
Affiliation:
Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, MI 48109, USA email rohitna@umich.edu
Peter Patzt
Affiliation:
Department of Mathematics, Purdue University, 150 North University Street, West Lafayette, IN 47904, USA email ppatzt@purdue.edu
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Abstract

We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of $\mathbf{SL}_{n}(\mathbb{Z})$. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group $\mathbf{SL}_{n}(K)$ for $K$ a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_{n}(\mathbb{Z})$.

Keywords

congruence subgroupsKoszul algebrasrepresentation stabilityBorel–Serre duality

MSC classification

Primary: 11F75: Cohomology of arithmetic groups 55N25: Homology with local coefficients, equivariant cohomology 20H05: Unimodular groups, congruence subgroups 19B14: Stability for linear groups 16S37: Quadratic and Koszul algebras
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 4 , April 2020 , pp. 822 - 861
Copyright
© The Authors 2020

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Footnotes

Jeremy Miller was supported in part by NSF grant DMS-1709726.

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