
Journal of Lie Theory 13 (2003), No. 1, 271278 Copyright Heldermann Verlag 2003 SousGroupes Elliptiques de Groupes Linéaires sur un Corps Valué Anne Parreau Laboratoire E. Picard, UMR 5580 du CNRS, Université Paul Sabatier, 31062 Toulouse, France [Abstractpdf] Let $n$ be a positive integer and $\mathbb{F}$ be a valuated field. We prove the following result: Let $\Gamma$ be a subgroup of $\mathrm{GL}_n(\mathbb{F})$ generated by a bounded subset, such that every element of $\Gamma$ belongs to a bounded subgroup. Then $\Gamma$ is bounded. \par This implies the following. Let $G$ be a connected reductive group over $\mathbb{F}$. Suppose that $\mathbb{F}$ is henselian (e.g. complete) and either that $G$ is almost split over $\mathbb{F}$, or that the valuation of $\mathbb{F}$ is discrete and $\mathbb{F}$ has perfect (e.g. finite) residue class field. Let $\Delta$ be its (extended) BruhatTits building. Let $x_0$ be any point in $\Delta$ and $\overline{\Delta}$ be the completion of $\Delta$. Let $\Gamma$ be a subgroup of $G$ generated by $S$ with $S.x_0$ bounded, such that every element of $\Gamma$ fixes a point in $\overline{\Delta}$, then $\Gamma$ has a global fixed point in $\overline{\Delta}$. [ Fulltextpdf (191 KB)] for subscribers only. 