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2021 On the top-dimensional cohomology groups of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$
Jeremy Miller, Peter Patzt, Andrew Putman
Geom. Topol. 25(2): 999-1058 (2021). DOI: 10.2140/gt.2021.25.999

Abstract

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 n2. We study the cohomology in this top degree n2. Let 𝒯n() denote the Tits building of SLn(). Lee and Szczarba conjectured that Hn2(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 Hn2(Γn(p))H̃n2(𝒯n()Γn(p)) is always surjective, but is only injective for p5. In particular, we completely calculate Hn2(Γn(5)) and improve known lower bounds for the ranks of Hn2(Γn(p)) for p5.

Citation

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Jeremy Miller. Peter Patzt. Andrew Putman. "On the top-dimensional cohomology groups of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$." Geom. Topol. 25 (2) 999 - 1058, 2021. https://doi.org/10.2140/gt.2021.25.999

Information

Received: 12 December 2019; Revised: 8 May 2020; Accepted: 6 June 2020; Published: 2021
First available in Project Euclid: 28 May 2021

Digital Object Identifier: 10.2140/gt.2021.25.999

Subjects:
Primary: 11F75

Keywords: Cohomology of arithmetic groups , congruence subgroups , Steinberg module

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.25 • No. 2 • 2021
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