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Bonjour,

Est ce qu'il existe une topologie naturelle sur l'ensemble des sous-groupes du groupe général linéaire ?

English translation: "is there a natural topology on the set of subgroups of the general linear group?"

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English translation: "is there a natural topology on the set of subgroups of the general linear group?" –  Yves Cornulier Feb 25 at 11:52
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If you refer to all subgroups, $GL_n(\mathbf{R})$ sounds like a not very natural setting. If you mean closed subgroups, there are at least two documented topologies (on the set of subgroups of any locally compact group): Chabauty topology and Vietoris topology. –  Yves Cornulier Feb 25 at 11:54
    
I was just writing an answer about the Chabauty topology when you commented ! I didn't know about the Vietoris topology though. –  Thomas Richard Feb 25 at 11:58
    
Both topologies are considered in Russian papers from the 80's, where the Chabauty topology is called "the compact topology". –  Yves Cornulier Feb 25 at 16:39
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1 Answer

English translation below

Si l'ensemble des sous-groupes fermés suffit, la topologie de Chabauty est une topologie naturelle sur l'ensemble des sous groupes fermés d'un groupe $G$ localement compacte. Pour des définitions précises et quelques résultats, voir par exemple:

http://www-fourier.ujf-grenoble.fr/~bkloeckn/papiers/ChabautyR2.pdf

If it is enough for you to restrict your self to closed subgroups, the Chabauty topology is a natural topology on the set of closed subgroups of a locally compact group $G$. Definitions and some results can be found here for instance:

http://www-fourier.ujf-grenoble.fr/~bkloeckn/papiers/ChabautyR2.pdf

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