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 $H^{\operatorname{vcd}}(\mathcal{G}(\mathfrak{O}); \mathbb{Q}) = 0$ if $\mathfrak{O}$ is Euclidean.