On the mod-$2$ cohomology of $\operatorname{SL}_3\bigl(\mathbb Z\bigl[\frac{1}{2},i\bigr]\bigr)$

Abstract

Let $\mathrm{\Gamma }={SL}_3\left(\mathbb{Z}\left[\frac{1}{2},i\right]\right)$, let $X$ be any mod-$2$ acyclic $\mathrm{\Gamma }$-CW complex on which $\mathrm{\Gamma }$ acts with finite stabilizers and let $X_s$ be the $2$-singular locus of $X$. We calculate the mod-$2$ cohomology of the Borel construction of $X_s$ with respect to the action of $\mathrm{\Gamma }$. This cohomology coincides with the mod-$2$ cohomology of $\mathrm{\Gamma }$ in cohomological degrees bigger than $8$ and the result is compatible with a conjecture of Quillen which predicts the structure of the cohomology ring $H^{\ast }\left(\mathrm{\Gamma };{\mathbb{F}}_2\right)$.