The strong approximation theorem and computing with linear groups

Abstract

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 $H\le \mathrm{SL}(n,\mathbb{Z})$ for $n\ge 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n,\mathbb{Q})$ for $n>2$.