Zariski density and computing in arithmetic groups

Abstract

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 <= 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.