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Borel-Serre proved that $\mathrm{SL}_n(\mathbb{Z})$ is a virtual dualitygroup of dimension $n \choose 2$ and the Steinberg module$\mathrm{St}_n(\mathbb{Q})$ is its dualizing module. This module is thetop-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 resolutionof length two of the $\mathrm{SL}_n(\mathbb{Z})$-module$\mathrm{St}_n(\mathbb{Q})$. We use this partial resolution to show thecodimension-2 rational cohomology group $H^{{n \choose 2}-2}(\mathrm{SL}_n(\mathbb{Z});\mathbb{Q})$ of $\mathrm{SL}_n(\mathbb{Z})$vanishes for $n \geq 3$. This resolves a case of a conjecture ofChurch-Farb-Putman. We also produce lower bounds for the codimension-1cohomology of certain congruence subgroups of $\mathrm{SL}_n(\mathbb{Z})$.

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