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 : 10.5802/aif.2822
Classification : 20P05, 20G20, 60B15
Keywords: transience, algebraic varieties, Zariski density, random matrix products, random walks, probability of return
Mot clés : propriétés génériques des groupes linéaires, marches aléatoires sur les groupes, produits de matrices aléatoires, sous-variétés des groupes algébriques linéaires
Aoun, Richard 1

1 Université Paris Sud 11 Laboratoire de Mathématiques Bâtiment 425 91405 Orsay (France) Département de Mathématiques Faculté des Sciences de l’Université Saint-Joseph Campus des Sciences et Technologies B.P. 11-514 Riad El Solh Beyrouth 1107 205 (Liban)
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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://www.numdam.org/articles/10.5802/aif.2822/

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