Consider a finitely generated Zariski dense subgroup 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 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 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.
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Richard Pink, Strong approximation for Zariski dense subgroups over arbitrary global fields. Comment. Math. Helv. 75 (2000), no. 4, pp. 608–643DOI 10.1007/S000140050142