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.

中文翻译：

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Quillen 猜想在 2 阶虚二次情况下的验证

我们在算术群 ${\rm SL}_2({\mathcal{O}}_{\mathbb{Q}(\sqrt{-m})}[ \frac{1}{2}]\thinspace)$，其中 ${\mathcal{O}}_{\mathbb{Q}(\sqrt{-m}\thinspace)}$ 是一个虚二次整数环。为了明确 Quillen 推测的上同调环上的自由模结构，我们计算了 ${\rm SL}_2({\mathbb{Z}}[\sqrt{-2}\thinspace][\ frac{1}{2}])$ 通过后一组的合并分解。