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

TL;DR

This paper verifies Quillen's conjecture for the mod 2 cohomology of certain arithmetic groups associated with imaginary quadratic fields, providing explicit computations for specific cases.

Contribution

It confirms Quillen's conjecture in the rank 2 imaginary quadratic case and explicitly computes the cohomology ring structure for a particular example.

Findings

Confirmed Quillen's conjecture for the mod 2 cohomology of SL_2 over imaginary quadratic rings.

Explicitly computed the cohomology of SL_2 over Z[√-2] using group decomposition.

Established the free module structure on the cohomology ring as conjectured.

Abstract

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