Strong approximation for Zariski dense subgroups over arbitrary global fields

Abstract

Consider a finitely generated Zariski dense subgroup $\Gamma$ of a connected simple algebraic group G over a global field F. An important aspect of strong approximation is the question of whether the closure of $\Gamma$ in the group of points of G with coefficients in a ring of partial adeles is open. We prove an essentially optimal result in this direction, based on the condition that $\Gamma$ is not discrete in that ambient group. There are no restrictions on the characteristic of F or the type of G, and simultaneous approximation in finitely many algebraic groups is also studied. Classification of finite simple groups is not used.