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Borel–Serre proved that SL_n(Z) is a virtual duality group of dimension (n choose 2) and the Steinberg module St_n(Q) is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to SL_n(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 SL_n(Z)-module St_n(Q). We use this partial resolution to show the codimension-2 rational cohomology group of SLn(Z) vanishes for n ≥ 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 SLn(Z).
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