On the 3-torsion in [formula omitted

Abstract

Let SL 4( Z) be the group of four by four integral matrices with determinant one. This group acts upon the top homology of the spherical Tits building of SL 4 over Q , i.e. the Steinberg module St 4 (see below, 1.2). The goal of this note is to prove the following: Theorem 1. The first homology group H 1(SL 4( Z), St 4) is a finite group of order a power of 2. This result was proved 18 years ago (Soulé, Thèse, University of Paris VII, 1979). At the time, I deduced from it that K 4( Z) is the direct sum of a finite 2-group and 0 or Z/3 . Rognes uses Theorem 1 in his proof that K 4( Z) vanishes (J. Rognes, K 4( Z) is the trivial group, Preprint, 1998).