Zariski density and computing in arithmetic groups

Detinko, A.; Flannery, D.; Hulpke, A.

doi:10.1090/mcom/3236

Zariski density and computing in arithmetic groups
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by A. Detinko, D. L. Flannery and A. Hulpke PDF

Math. Comp. 87 (2018), 967-986 Request permission

Abstract:

For $n > 2$, let $\Gamma _n$ denote either $\mathrm {SL}(n, \mathbb {Z})$ or $\mathrm {Sp}(n, \mathbb {Z})$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq \Gamma _n$. 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 $\Gamma _n$. 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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