On the top-dimensional cohomology groups of congruence subgroups of SL(n, ℤ)

Miller, Jeremy; Patzt, Peter; Putman, Andrew

doi:10.2140/gt.2021.25.999

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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

DOI: 10.2140/gt.2021.25.999

Abstract

Let $Γ_{n} (p)$ be the level- $p$ principal congruence subgroup of ${SL}_{n} (ℤ)$ . Borel and Serre proved that the cohomology of $Γ_{n} (p)$ vanishes above degree $(\binom{n}{2})$ . We study the cohomology in this top degree $(\binom{n}{2})$ . Let $𝒯_{n} (ℚ)$ denote the Tits building of ${SL}_{n} (ℚ)$ . Lee and Szczarba conjectured that $H^{(\binom{n}{2})} (Γ_{n} (p))$ is isomorphic to ${\tilde{H}}_{n - 2} (𝒯_{n} (ℚ) ∕ Γ_{n} (p))$ and proved that this holds for $p = 3$ . We partially prove and partially disprove this conjecture by showing that a natural map $H^{(\binom{n}{2})} (Γ_{n} (p)) \to {\tilde{H}}_{n - 2} (𝒯_{n} (ℚ) ∕ Γ_{n} (p))$ is always surjective, but is only injective for $p \leq 5$ . In particular, we completely calculate $H^{(\binom{n}{2})} (Γ_{n} (5))$ and improve known lower bounds for the ranks of $H^{(\binom{n}{2})} (Γ_{n} (p))$ for $p \geq 5$ .

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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

Volume 25, issue 2 (2021)