The codimension-one cohomology of SL_n Z

TL;DR

This paper proves the vanishing of the top-degree rational cohomology of SL_n Z and GL_n Z, using a new topological proof of a presentation of the Steinberg module based on integral apartment classes.

Contribution

It introduces a new topological proof for the presentation of the Steinberg module of SL_n Z, leading to vanishing theorems for cohomology.

Findings

Proves H^{d-1}(SL_n Z; Q) = 0 for d = n-choose-2

Establishes similar vanishing for GL_n Z cohomology

Provides a new topological proof of the Steinberg module presentation

Abstract

We prove that H^{d-1}(SL_n Z; Q) = 0, where d = n-choose-2 is the cohomological dimension of SL_n 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 SL_2 Z. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.