Algorithms for experimenting with Zariski dense subgroups

TL;DR

This paper develops algorithms to describe and compute congruence images of Zariski dense subgroups in special linear groups, with implementations in GAP for experimental exploration.

Contribution

It introduces new methods for describing all congruence images of finitely generated Zariski dense groups, including efficient algorithms for odd prime degrees and a universal approach for groups containing a transvection.

Findings

Algorithms successfully implemented in GAP.

Efficient computation for odd prime degrees.

Method applicable to all n with known transvection.

Abstract

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