The strong approximation theorem and computing with linear groups

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.