On the Mod-2 Cohomology of SL 3 (z[ 1 2 , i])

Abstract

Let $\Gamma$ = SL 3 (Z[ 1 2 , i]), let X be any mod-2 acyclic $\Gamma$-CW complex on which $\Gamma$ acts with finite stabilizers and let Xs be the 2-singular locus of X. We calculate the mod-2 cohomology of the Borel constructon of Xs with respect to the action of $\Gamma$. This cohomology coincides with the mod-2 cohomology of $\Gamma$ in cohomological degrees bigger than 8 and the result is compatible with a conjecture of Quillen which predicts the strucure of the cohomology ring H * ($\Gamma$; Z/2).