Steinberg module and top-dimensional cohomology for arithmetic Chevalley groups
Miguel Gentili
Abstract
We prove that for a Chevalley--Demazure group scheme G of type G2, F4 or E8 and a Euclidean ring O with field of fractions K, the Steinberg module of G(K) is generated by integral apartment classes. As a consequence, the top-dimensional rational cohomology of G(O) vanishes if O is a ring of integers, extending results of Tóth and of Brück--Santos Rego--Sroka.
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