Abstract
Borel–Serre proved that SLn(Z) is a virtual duality group of dimension (n2) and the Steinberg module Stn(Q) is its dualizing module. This module is the top-dimensional homology group of the Tits building associated to SLn(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 SLn(Z)-module Stn(Q). We use this partial resolution to show the codimension-2 rational cohomology group H(n2)−2(SLn(Z);Q) 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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