On the codimension-two cohomology of ${\mathrm{SL}}_n(\mathbb{Z})$

Abstract

Borel-Serre proved that ${\mathrm{SL}}_n(\mathbb{Z})$ is a virtual duality group of dimension $\left(\genfrac{}{}{0ex}{}{n}{2}\right)$ and the Steinberg module ${\mathrm{St}}_n(\mathbb{Q})$ is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to ${\mathrm{SL}}_n(\mathbb{Q})$. We determine the "relations among the relations" of this Steinberg module. That is, we construct an explicit partial resolution of length two of the ${\mathrm{SL}}_n(\mathbb{Z})$-module ${\mathrm{St}}_n(\mathbb{Q})$. We use this partial resolution to show the codimension-2 rational cohomology group $H^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)-2}({\mathrm{SL}}_n(\mathbb{Z});\mathbb{Q})$ of ${\mathrm{SL}}_n(\mathbb{Z})$ vanishes for $n\ge 3$. This resolves a case of a conjecture of Church-Farb-Putman. We also produce lower bounds for the codimension-1 cohomology of certain congruence subgroups of ${\mathrm{SL}}_n(\mathbb{Z})$.

arXiv comment65 pages, 13 figures; v3: changes according to the referee's suggestions, to appear in Adv. Math

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