On the codimension-two cohomology of SLn(Z)

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....

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2024-08, Vol.451, Article 109795
Main Authors: Brück, Benjamin, Miller, Jeremy, Patzt, Peter, Sroka, Robin J., Wilson, Jennifer C.H.
Format: Article
Language:English
Subjects:
Group cohomology
Special linear groups
Steinberg modules
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Summary: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).
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2024.109795