Zariski density and computing in arithmetic groups

For n > 2, let Gamma _n denote either SL(n, {Z}) or Sp(n, {Z}). We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group H\leq \Gamma _n. This forms the main component of our methods for computing with such arithmetic groups H. More generally, we provide algorithms for computing with Zariski dense groups in Gamma _n. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups