Abstract
We prove that , where is the cohomological dimension of , and similarly for . We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group . These theorems are derived from a presentation of the Steinberg module for whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of . This presentation was originally constructed by Bykovskiĭ. We give a new topological proof of it.
Citation
Thomas Church. Andrew Putman. "The codimension-one cohomology of $\mathrm{SL}_n \mathbb{Z}$." Geom. Topol. 21 (2) 999 - 1032, 2017. https://doi.org/10.2140/gt.2017.21.999
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