Volume 21, issue 2 (2017)

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The codimension-one cohomology of $\mathrm{SL}_n \mathbb{Z}$

Thomas Church and Andrew Putman

Geometry & Topology 21 (2017) 999–1032
Abstract

We prove that ${H}^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)-1}\left({SL}_{n}\phantom{\rule{0.3em}{0ex}}ℤ;ℚ\right)=0$, where $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ is the cohomological dimension of ${SL}_{n}\phantom{\rule{0.3em}{0ex}}ℤ$, and similarly for ${GL}_{n}\phantom{\rule{0.3em}{0ex}}ℤ$. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group ${GL}_{n}$. These theorems are derived from a presentation of the Steinberg module for ${SL}_{n}\phantom{\rule{0.3em}{0ex}}ℤ$ whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of ${SL}_{2}\phantom{\rule{0.3em}{0ex}}ℤ$. This presentation was originally constructed by Bykovskiĭ. We give a new topological proof of it.

Keywords
cohomology of arithmetic groups, Steinberg module, partial bases
Primary: 11F75