# The strong approximation theorem and computing with linear groups

## TL;DR

This paper develops algorithms to compute all congruence quotients of finitely generated Zariski dense subgroups of special linear groups, providing a computational realization of the strong approximation theorem.

## Contribution

It introduces a method to explicitly compute congruence quotients for Zariski dense subgroups of SL(n, Z) and SL(n, Q), extending the practical application of the strong approximation theorem.

## Key findings

- Algorithms successfully compute all congruence quotients for specified groups.
- The approach applies to groups in SL(n, Z) for n ≥ 2 and in SL(n, Q) for n > 2.
- Provides a computational framework for strong approximation in linear groups.

## Abstract

We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group $H \leq \mathrm{SL}(n, \mathbb{Z})$ for $n \geq 2$. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of $\mathrm{SL}(n, \mathbb{Q})$ for $n > 2$.