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Verification of the Quillen conjecture in the rank 2 imaginary quadratic case. (English) Zbl 1440.11088

Summary: We confirm a conjecture of Quillen in the case of the \(\operatorname{mod} 2\) cohomology of arithmetic groups \(\mathrm{SL}_2(\mathcal{O}_{\mathbb{Q}(\sqrt{-m})}[\frac{1}{2}])\), where \(\mathcal{O}_{\mathbb{Q}(\sqrt{-m})}\) is an imaginary quadratic ring of integers. To make explicit the free module structure on the cohomology ring conjectured by Quillen, we compute the \(\operatorname{mod} 2\) cohomology of \(\text{SL}_2(\mathbb{Z}[\sqrt{-2}][\frac{1}{2}])\) via the amalgamated decomposition of the latter group.

MSC:

11F75 Cohomology of arithmetic groups
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