On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ)
Geometry & topology, Tome 25 (2021) no. 2, pp. 999-1058.

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Let Γn(p) be the level-p principal congruence subgroup of SLn(). Borel and Serre proved that the cohomology of Γn(p) vanishes above degree n 2 . We study the cohomology in this top degree n 2 . Let 𝒯n() denote the Tits building of SLn(). Lee and Szczarba conjectured that Hn 2 (Γn(p)) is isomorphic to H̃n2(𝒯n()Γn(p)) and proved that this holds for p = 3. We partially prove and partially disprove this conjecture by showing that a natural map Hn 2 (Γn(p)) H̃n2(𝒯n()Γn(p)) is always surjective, but is only injective for p 5. In particular, we completely calculate Hn 2 (Γn(5)) and improve known lower bounds for the ranks of Hn 2 (Γn(p)) for p 5.

DOI : 10.2140/gt.2021.25.999
Classification : 11F75
Keywords: congruence subgroups, Steinberg module, cohomology of arithmetic groups

Miller, Jeremy 1 ; Patzt, Peter 1 ; Putman, Andrew 2

1 Department of Mathematics, Purdue University, West Lafayette, IN, United States
2 Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States
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Miller, Jeremy; Patzt, Peter; Putman, Andrew. On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ). Geometry & topology, Tome 25 (2021) no. 2, pp. 999-1058. doi : 10.2140/gt.2021.25.999. http://geodesic.mathdoc.fr/articles/10.2140/gt.2021.25.999/

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