Strong Approximation for Zariski-dense Subgroups of Semi-Simple Algebraic Groups

(1.1) THEOREM. Let k be an algebraically closed field of characteristic different from 2 and 3, and G an almost simple, connected and simply connected algebraic group defined over k. Let F be a finitely generated Zariski dense subgroup of G(k) and A the subring of k generated by the traces tr Ad y, y e P. Then there exist b e A, a subgroup F' of finite index in F, and a structure GAb of a group scheme over Ab on G such that F' C GA(Ab) and F' is dense in GAlA b)A