Algorithms for experimenting with Zariski dense subgroups

Alla Detinko; Dane Flannery; Alexander Hulpke

doi:10.1080/10586458.2018.1466217

Algorithms for experimenting with Zariski dense subgroups

Alla Detinko, Dane Flannery, Alexander Hulpke

School of Computer Science

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

4 Citations (Scopus)

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Abstract

We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL (n,ℤ). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n;if n=2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.

Original language	English
Number of pages	10
Journal	Experimental Mathematics
Volume	Latest Articles
Early online date	4 Jun 2018
DOIs	https://doi.org/10.1080/10586458.2018.1466217
Publication status	E-pub ahead of print - 4 Jun 2018

Keywords

Algorithm
Zariski dense
Congruence subgroup
Strong approximation

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10.1080/10586458.2018.1466217

Detinko_2018_EM_Zariski_AAM
© 2018 Taylor & Francis. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1080/10586458.2018.1466217
Accepted author manuscript, 214 KB

https://arxiv.org/abs/1711.02147

Cite this

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abstract = "We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL (n,ℤ). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n;if n=2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.",

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AU - Detinko, Alla

AU - Flannery, Dane

AU - Hulpke, Alexander

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N2 - We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL (n,ℤ). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n;if n=2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.

AB - We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL (n,ℤ). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n;if n=2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.

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KW - Zariski dense

KW - Congruence subgroup

KW - Strong approximation

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VL - Latest Articles

JO - Experimental Mathematics

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ER -

Algorithms for experimenting with Zariski dense subgroups

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Keywords

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