Transience of algebraic varieties in linear groups - applications to generic Zariski density

Aoun, Richard

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 $S{L}_2\left(ℝ\right)$ 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 $ℝ$.

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 $S{L}_2\left(ℝ\right)$ 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.

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

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