Zariski density and computing with $S$-integral groups

Abstract: We generalize our methodology for computing with Zariski dense subgroups of $\mathrm{SL}(n, \mathbb{Z})$ and $\mathrm{Sp}(n, \mathbb{Z})$, to accommodate input dense subgroups $H$ of $\mathrm{SL}(n, \mathbb{Q})$ and $\mathrm{Sp}(n, \mathbb{Q})$. A key task, backgrounded by the Strong Approximation theorem, is computing a minimal congruence overgroup of $H$. Once we have this overgroup, we may describe all congruence quotients of $H$. The case $n=2$ receives particular attention.