Transience of algebraic varieties in linear groups - applications to generic Zariski density
Annales de l'Institut Fourier, Volume 63 (2013) no. 5, pp. 2049-2080.

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.

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.

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, Volume 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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