The codimension-one cohomology of SLnℤ

Church, Thomas¹ ; Putman, Andrew²

¹ Department of Mathematics, Stanford University, 450 Serra Mall, Stanford, CA 94305, United States
² Department of Mathematics, University of Notre Dame, 255 Hurley, Notre Dame, IN 46556, United States

Geometry & topology, Tome 21 (2017) no. 2, pp. 999-1032.

Résumé

We prove that $H^{\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)-1}\left({SL}_n\mathbb{Z};\mathbb{Q}\right)=0$, where $\left(\genfrac{}{}{0.0pt}{}{n}{2}\right)$ is the cohomological dimension of ${SL}_n\mathbb{Z}$, and similarly for ${GL}_n\mathbb{Z}$. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group ${GL}_n$. These theorems are derived from a presentation of the Steinberg module for ${SL}_n\mathbb{Z}$ whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of ${SL}_2\mathbb{Z}$. This presentation was originally constructed by Bykovskiĭ. We give a new topological proof of it.

Détail

DOI : 10.2140/gt.2017.21.999

Classification : 11F75
Keywords: cohomology of arithmetic groups, Steinberg module, partial bases

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