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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
The codimension-one cohomology of $\mathrm{SL}_n \mathbb{Z}$

Thomas Church and Andrew Putman

Geometry & Topology 21 (2017) 999–1032
Abstract

We prove that Hn 2 1(SLn; ) = 0, where n 2 is the cohomological dimension of SLn, and similarly for GLn. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GLn. These theorems are derived from a presentation of the Steinberg module for SLn whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of SL2. This presentation was originally constructed by Bykovskiĭ. We give a new topological proof of it.

Keywords
cohomology of arithmetic groups, Steinberg module, partial bases
Mathematical Subject Classification 2010
Primary: 11F75
References
Publication
Received: 4 August 2015
Revised: 17 April 2016
Accepted: 13 July 2016
Published: 17 March 2017
Proposed: Ronald Stern
Seconded: Mark Behrens, András I. Stipsicz
Authors
Thomas Church
Department of Mathematics
Stanford University
450 Serra Mall
Stanford, CA 94305
United States
http://math.stanford.edu/~church/
Andrew Putman
Department of Mathematics
University of Notre Dame
255 Hurley
Notre Dame, IN 46556
United States
http://www3.nd.edu/~andyp/