Presentations of parabolics in some elementary Chevalley-Demazure groups

14 Feb 2018  ·  Rego Yuri Santos ·

Given a universal elementary Chevalley-Demazure group $E_\Phi^{sc}(R)$ for which its (standard) parabolic subgroups are finitely generated, we consider the problem of classifying which parabolics $P(R) \subset E_\Phi^{sc}(R)$ are finitely presented. We show that, under mild assumptions, this is equivalent to the finite presentability of a suitable retract of $P$ which contains the Levi factor... If the base ring $R$ is a Dedekind domain of arithmetic type, we combine our results with well-known theorems due to Borel-Serre, Abels, Behr and Bux to give a partial classification of finitely presentable $S$-arithmetic subgroups of parabolics in split reductive linear algebraic groups. read more

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Group Theory