On the top-dimensional cohomology of arithmetic Chevalley groups

Abstract

Let K \mathbb {K} be a number field with ring of integers O \mathfrak {O} and let G \mathcal {G} be a Chevalley group scheme not of type E 8 \mathtt {E}_8 , F 4 \mathtt {F}_4 or G 2 \mathtt {G}_2 . We use the theory of Tits buildings and a result of Tóth on Steinberg modules to prove that H v c d ( G ( O ) ; Q ) = 0 H^{{vcd}}(\mathcal {G}(\mathfrak {O}); \mathbb {Q}) = 0 if O \mathfrak {O} is Euclidean.