Zariski Density and Computing in Arithmetic 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 $H\le \Gamma$. 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.

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