Zariski density and computing in arithmetic groups

A. Detinko; D. L. Flannery; A. Hulpke

doi:10.1090/mcom/3236

Zariski density and computing in arithmetic groups

A. Detinko, D. L. Flannery, A. Hulpke

School of Computer Science

Research output: Contribution to journal › Article › peer-review

Abstract

For n > 2, let Γ_n 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 ≤ Γ_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 Γ_n. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

Original language	English
Pages (from-to)	967-986
Journal	Mathematics of Computation
Volume	87
Issue number	310
Early online date	7 Aug 2017
DOIs	https://doi.org/10.1090/mcom/3236
Publication status	Published - Mar 2018

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title = "Zariski density and computing in arithmetic groups",

abstract = "For n > 2, let Γn 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 ≤ Γn. 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 Γn. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.",

author = "A. Detinko and Flannery, \{D. L.\} and A. Hulpke",

note = "The first and second authors received support from the Irish Research Council (grants {\textquoteleft}MatGpAlg{\textquoteright} and {\textquoteleft}MatGroups{\textquoteright}) and Science Foundation Ireland (grant 11/RFP.1/MTH/3212). The first author is also funded by a Marie Sklodowska-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.",

year = "2018",

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doi = "10.1090/mcom/3236",

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AU - Detinko, A.

AU - Flannery, D. L.

AU - Hulpke, A.

N1 - The first and second authors received support from the Irish Research Council (grants ‘MatGpAlg’ and ‘MatGroups’) and Science Foundation Ireland (grant 11/RFP.1/MTH/3212). The first author is also funded by a Marie Sklodowska-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.

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N2 - For n > 2, let Γn 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 ≤ Γn. 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 Γn. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

AB - For n > 2, let Γn 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 ≤ Γn. 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 Γn. 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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VL - 87

SP - 967

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JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 310

ER -

Zariski density and computing in arithmetic groups

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