Volume 25, issue 2 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
This article is available for purchase or by subscription. See below.
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

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.

PDF Access Denied

We have not been able to recognize your IP address as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

Please contact your institution's librarian suggesting a subscription, for example by using our journal-recom­mendation form. Or, visit our subscription page for instructions on purchasing a subscription.

You may also contact us at contact@msp.org
or by using our contact form.

Or, you may purchase this single article for USD 40.00:

congruence subgroups, Steinberg module, cohomology of arithmetic groups
Mathematical Subject Classification 2010
Primary: 11F75
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
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