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arXiv · 1806.11131

Stability in the high-dimensional cohomology of congruence subgroups

Jeremy Miller·Rohit Nagpal·Peter Patzt
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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-Putman which we make more precise. Our methods involve proving several 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-Sam's homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church-Putman's homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_n(\mathbb{Z})$.

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BibTeXRIS

Jeremy Miller, Rohit Nagpal, Peter Patzt. 2018-06-28. Stability in the high-dimensional cohomology of congruence subgroups. https://doi.org/10.1112/s0010437x20007046

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