Topological generation of linear algebraic groups I

Spencer Gerhardt

Published 2018-01-30Version 1

Let $C_1,...,C_e$ be noncentral conjugacy classes of the algebraic group $G=SL_n(k)$ defined over a sufficiently large field $k$, and let $\Omega:=C_1\times ...\times C_e$. This paper determines necessary and sufficient conditions for the existence of a tuple $(x_1,...,x_e)\in\Omega$ such that $\langle x_1,...,x_e\rangle$ 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.