Transience of algebraic varieties in linear groups - applications to generic Zariski density
[Transience des variétés algébriques dans les groupes linéaires - applications à la généricité de la notion de densité Zariski]
Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 2049-2080.

Nous étudions la transience des variétés algébriques dans les groupes linéaires. En particulier, nous montrons qu’une marche aléatoire sur un sous-groupe non élémentaire de SL 2 () évite toute sous-variété algébrique propre avec une probabilité convergeant vers 1 de façon exponentielle. Nous étudions aussi le cas où la marche aléatoire vit dans un sous-groupe Zariski dense du groupe des points réels d’un groupe algébrique semi-simple, défini et déployé sur .

Nous utilisons ces résultats pour montrer qu’un sous-groupe aléatoire (en un sens à préciser) d’un groupe algébrique est Zariski dense.

We study the transience of algebraic varieties in linear groups. In particular, we show that a “non elementary” random walk in SL 2 () escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk takes place in the real points of a semisimple split algebraic group and show such a result for a wide family of random walks.

As an application, we prove that generic subgroups (in some sense) of linear groups are Zariski dense.

DOI : https://doi.org/10.5802/aif.2822
Classification : 20P05,  20G20,  60B15
Mots clés : propriétés génériques des groupes linéaires, marches aléatoires sur les groupes, produits de matrices aléatoires, sous-variétés des groupes algébriques linéaires
@article{AIF_2013__63_5_2049_0,
     author = {Aoun, Richard},
     title = {Transience of algebraic varieties in linear groups - applications to generic Zariski density},
     journal = {Annales de l'Institut Fourier},
     pages = {2049--2080},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {63},
     number = {5},
     year = {2013},
     doi = {10.5802/aif.2822},
     mrnumber = {3203113},
     zbl = {06284540},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2822/}
}
Aoun, Richard. Transience of algebraic varieties in linear groups - applications to generic Zariski density. Annales de l'Institut Fourier, Tome 63 (2013) no. 5, pp. 2049-2080. doi : 10.5802/aif.2822. http://archive.numdam.org/articles/10.5802/aif.2822/

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