Topological generation of special linear groups

Abstract

Let $C_1,\dots ,C_e$ be noncentral conjugacy classes of the algebraic group $G=S{L}_n(k)$ defined over a sufficiently large field $k$, and let $\Omega :=C_1\times \dots \times C_e$. This paper determines necessary and sufficient conditions for the existence of a tuple $(x_1,\dots ,x_e)\in \Omega$ such that $⟨x_1,\dots ,x_e⟩$ is Zariski dense in $G$. As a consequence, a new result concerning generic stabilizers in linear representations of algebraic groups is proved, and existing results on random $(r,s)$-generation of finite groups of Lie type are strengthened.

33 pages; to appear in Journal of Algebra