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.

KEYWORDS:

• algorithm
• Zariski dense
• congruence subgroup
• strong approximation

2010 AMS SUBJECT CLASSIFICATION:

• 20-04
• 20G15
• 20H25
• 68W30

Acknowledgments

We thank Mathematisches Forschungsinstitut Oberwolfach and the International Centre for Mathematical Sciences, Edinburgh, for hosting visits in 2017 under their Research-in-Pairs and Research in Groups programmes, respectively.

Additional information

Funding

Our work was additionally supported by a Marie Skłodowska-Curie Individual Fellowship grant (Horizon 2020, EU Framework Programme for Research and Innovation), and Simons Foundation Collaboration Grant 244502.