#### Vol. 1, No. 4, 2019

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

### Hans-Werner Henn

Vol. 1 (2019), No. 4, 539–560
##### Abstract

Let $\Gamma ={SL}_{3}\left(ℤ\left[\frac{1}{2},i\right]\right)$, 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 ${H}^{\ast }\left(\Gamma ;{\mathbb{F}}_{2}\right)$.

##### Keywords
cohomology of special and general linear groups, Quillen conjecture
Primary: 20G10
Secondary: 55R40