Algorithms for Experimenting with Zariski Dense Subgroups

We give a method to describe all congruence images of a finitely generated Zariski dense group H≤SL(n,Z). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n; if n = 2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.