Volume 25, issue 2 (2021)

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On the top-dimensional cohomology groups of congruence subgroups of $\mathrm{SL}(n,\mathbb{Z})$

Jeremy Miller, Peter Patzt and Andrew Putman

Geometry & Topology 25 (2021) 999–1058
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 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.

Keywords
congruence subgroups, Steinberg module, cohomology of arithmetic groups
Mathematical Subject Classification 2010
Primary: 11F75
References
Publication
Received: 12 December 2019
Revised: 8 May 2020
Accepted: 6 June 2020
Published: 27 April 2021
Proposed: Mladen Bestvina
Seconded: Haynes R Miller, Bruce Kleiner
Authors
Jeremy Miller
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Peter Patzt
Department of Mathematics
Purdue University
West Lafayette, IN
United States
Andrew Putman
Department of Mathematics
University of Notre Dame
Notre Dame, IN
United States