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Journal of Lie Theory 13 (2003), No. 1, 271--278
Copyright Heldermann Verlag 2003

Sous-Groupes Elliptiques de Groupes Linéaires sur un Corps Valué

Anne Parreau
Laboratoire E. Picard, UMR 5580 du CNRS, Université Paul Sabatier, 31062 Toulouse, France


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) Bruhat-Tits 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 $\overline{\Delta}$.

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