Stability in the high-dimensional cohomology of congruence subgroups

TL;DR

This paper establishes a stability result for the cohomology of certain congruence subgroups of SL_n, using finiteness properties of the Steinberg module, and refines existing homological vanishing theorems.

Contribution

It provides a new stability theorem for the cohomology of congruence subgroups and introduces finiteness properties of the Steinberg module, along with refined vanishing results.

Findings

Proved a representation stability result for the codimension-one cohomology of the level three congruence subgroup of SL_n.

Established finiteness properties of the Steinberg module for SL_n over fields.

Provided a new proof and an integral refinement of existing homological vanishing theorems.

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 $\mathbf{SL}_n(\mathbb{Z})$.