Journal of Lie Theory

no. 1
Sous-Groupes Elliptiques de Groupes Linéaires sur un Corps Valué
Journal of Lie Theory, Volume 13 (2003) no. 1, pp. 271-278
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 $\overline{\Delta}$.
DOI: 10.5802/jolt.302
Classification: 20G25
Keywords: elliptic elements, local fields, general linear groups, bounded subgroups 
@article{JOLT_2003_13_1_a15,
     author = {A. Parreau},
     title = {Sous-Groupes {Elliptiques} de {Groupes} {Lin\'eaires} sur un {Corps} {Valu\'e}},
     journal = {Journal of Lie Theory},
     pages = {271--278},
     year = {2003},
     volume = {13},
     number = {1},
     doi = {10.5802/jolt.302},
     zbl = {1021.20035},
     url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.302/}
}
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A. Parreau. Sous-Groupes Elliptiques de Groupes Linéaires sur un Corps Valué. Journal of Lie Theory, Volume 13 (2003) no. 1, pp. 271-278. doi: 10.5802/jolt.302

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