Algorithms for experimenting with Zariski dense matrix groups over number fields

Let $\mathbb{P}$ be an algebraic number field. We provide a computational analog of the strong approximation theorem for finitely generated Zariski dense groups $H\le \mathrm{S}\mathrm{L}(n,\mathbb{P})$, $n$ 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 $\mathbb{P}$ such that $H\le \mathrm{S}\mathrm{L}(n,R)$. 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.