Let
$\Gamma ={SL}_{3}\left(\mathbb{Z}\left[\frac{1}{2},i\right]\right)$, let
$X$ be any
mod$2$ acyclic
$\Gamma $CW complex on
which
$\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
$\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 structure of the
cohomology ring
${H}^{\ast}\left(\Gamma ;{\mathbb{F}}_{2}\right)$.
