Algorithms for experimenting with Zariski dense subgroups

Abstract

We give a method to describe all congruence images of a finitely generated Zariski dense group $H\le \mathrm{SL}(n,\mathbb{Z})$. 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.

References in corpus (2)

• Zariski Density and Computing in Arithmetic Groups
• Generic Phenomena in Groups -- Some Answers and Many Questions

Cited by in corpus (3)

• Experimenting with symplectic hypergeometric monodromy groups
• The strong approximation theorem and computing with linear groups
• Algorithms for arithmetic groups with the congruence subgroup property