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Zariski density and computing in arithmetic groups


Authors: A. Detinko, D. L. Flannery and A. Hulpke
Journal: Math. Comp. 87 (2018), 967-986
MSC (2010): Primary 20H05, 20B40
DOI: https://doi.org/10.1090/mcom/3236
Published electronically: August 7, 2017
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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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Additional Information

A. Detinko
Affiliation: School of Computer Science, University of St Andrews, North Haugh, St Andrews, KY16 9SX, United Kingdom

D. L. Flannery
Affiliation: School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, University Road, Galway, Ireland

A. Hulpke
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523-1874

DOI: https://doi.org/10.1090/mcom/3236
Received by editor(s): June 14, 2016
Received by editor(s) in revised form: October 5, 2016, and October 26, 2016
Published electronically: August 7, 2017
Additional Notes: The first and second authors received support from the Irish Research Council (grants ‘MatGpAlg’ and ‘MatGroups’) and Science Foundation Ireland (grant 11/RFP.1/\abk MTH/3212). The first author is also funded by a Marie Skłodowska-Curie Individual Fellowship grant under Horizon 2020 (EU Framework Programme for Research and Innovation).
The third author was supported by Simons Foundation Collaboration Grant 244502
Article copyright: © Copyright 2017 American Mathematical Society

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