Homology, Homotopy and Applications

Volume 22 (2020)

Number 2

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

Pages: 265 – 278

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n2.a17


Bui Anh Tuan (Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam)

Alexander D. Rahm (Laboratoire de mathématiques, Université de la Polynésie française, Faaa, French Polynesia)


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.


cohomology of arithmetic groups

2010 Mathematics Subject Classification


Received 18 December 2017

Received revised 8 November 2019

Published 6 May 2020