The codimension-one cohomology of SL(n)Z

We prove that H ((n)(2))(-1)SL(n)Z;Q = 0, where ((n)(2)) is the cohomological dimension of SLn Z, and similarly for GL(n) 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)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.