Transience of algebraic varieties in linear groups and application to generic Zariski density

Abstract

We study the transience of algebraic varieties in linear groups. In particular, we show that a "non elementary" random walk in SL_2(R) escapes exponentially fast from every proper algebraic subvariety. We also treat the case where the random walk is on the real points of a semi-simple 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.

References in corpus (3)

• Random subgroups of linear groups are free
• A Strong Tits Alternative
• Expansion in $S{L}_d(O_K/I)$, $I$ square-free

Cited by in corpus (1)

• A spectral gap theorem in simple Lie groups