Verification of the Quillen conjecture in the rank 2 imaginary quadratic case

We confirm a conjecture of Quillen in the case of the $\operatorname{mod} 2$ cohomology of arithmetic groups ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\, )}[\frac{1}{2}])$, 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 $\mathrm{SL}_2(\mathbb{Z}[\sqrt{-2}\,][\frac{1}{2}])$ via the amalgamated decomposition of the latter group.

Keywords

cohomology of arithmetic groups

2010 Mathematics Subject Classification

11F75

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Received 18 December 2017

Received revised 8 November 2019

Published 6 May 2020