The codimension-one cohomology of SLnZ

We prove that H( n 2)(SLn Z;Q) = 0, where ( n 2 ) is the cohomological dimension of SLn Z, and similarly for GLn Z. 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 Z whose generators are integral apartment classes, generalizing Manin’s presentation for the Steinberg module of SL2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.