On the top-dimensional cohomology of arithmetic Chevalley groups

Abstract

Let $\mathbb{K}$ be a number field with ring of integers $\mathfrak{O}$ and let $\mathcal{G}$ be a Chevalley group scheme not of type ${\mathtt{E}}_8$, ${\mathtt{F}}_4$ or ${\mathtt{G}}_2$. We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that $H^{\mathrm{vcd}}(\mathcal{G}(\mathfrak{O});\mathbb{Q})=0$ if $\mathfrak{O}$ is Euclidean.

arXiv comment8 pages; v2: small changes according to referee's suggestions; to appear in Proc. Amer. Math. Soc

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Cited by in corpus (2)

• Apartment classes of integral symplectic groups
• Top-degree rational cohomology in the symplectic group of a number ring