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 $n > 2$.