Algorithms for experimenting with Zariski dense matrix groups over number fields

Let P be an algebraic number field. We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense subgroups H of SL(n, P), n prime. That is, we present algorithms to find the set of congruence quotients of H modulo all maximal ideals of a finitely generated subring R of P such that H <SL(n, R) . The algorithms have been implemented in GAP. Potential applications are illustrated by a range of experiments in degree 2, with a special focus on Bianchi groups.