Zariski Density and Computing in Arithmetic Groups

TL;DR

This paper presents practical algorithms for computing levels of principal congruence subgroups and Zariski dense groups within arithmetic groups like SL(n,Z) and Sp(n,Z), enabling advanced computations in linear group theory.

Contribution

It introduces new algorithms and a GAP implementation for computing with Zariski dense subgroups and principal congruence levels in arithmetic groups, advancing computational methods in this area.

Findings

Algorithms successfully compute levels of principal congruence subgroups.

GAP implementation enables solving recent problems in linear group computations.

Methods applicable to important classes of linear groups.

Abstract

For $n > 2$, let $\Gamma$ 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 $H\leq \Gamma$. 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 $\Gamma$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.