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

Abstract

We prove that $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)-1}\left({SL}_n\mathbb{Z};\mathbb{Q}\right)=0$, where $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ is the cohomological dimension of ${SL}_n\mathbb{Z}$, and similarly for ${GL}_n\mathbb{Z}$. 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 ${SL}_n\mathbb{Z}$ whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of ${SL}_2\mathbb{Z}$. This presentation was originally constructed by Bykovskiĭ. We give a new topological proof of it.