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

Bui Anh Tuan ; Alexander D. Rahm

Homology, homotopy, and applications, Tome 22 (2020) no. 2, pp. 265-278.

Résumé

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.

Détail

DOI : 10.4310/HHA.2020.v22.n2.a17

Classification : 11F75
Keywords: cohomology of arithmetic groups