Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
We prove that for a Chevalley--Demazure group scheme $\mathcal{G}$ of type $G_2$, $F_4$ or $E_8$ and a Euclidean ring $\mathcal{O}$ with field of fractions $K$, the Steinberg module of $\mathcal{G}(K)$ is generated by integral apartment classes. As a consequence, the top-dimensional rational cohomology of $\mathcal{G}(\mathcal{O})$ vanishes if $\mathcal{O}$ is a ring of integers, extending results of Tóth and of Brück--Santos Rego--Sroka.
math.GR↗