Presentations of parabolics in some elementary Chevalley-Demazure groups

TL;DR

This paper investigates the conditions under which parabolic subgroups of elementary Chevalley-Demazure groups are finitely presented, linking this property to their retracts and applying results to arithmetic groups over Dedekind domains.

Contribution

It establishes a criterion for finite presentability of parabolics in Chevalley-Demazure groups and provides a partial classification for arithmetic subgroups over Dedekind domains.

Findings

Finite presentability of parabolics is equivalent to that of certain retracts.

Results apply to S-arithmetic subgroups over Dedekind domains.

Partial classification of finitely presented S-arithmetic subgroups in split reductive groups.

Abstract

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.