Freeness and S-arithmeticity of rational Möbius groups

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

Freeness and S-arithmeticity of rational Möbius groups

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

Abstract

We initiate a new, computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, Q). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, R) for a localization R = Z[¹b^] of Z. We prove that a Möbius group G ≤ SL(2, R) is not free by showing that it has finite index in SL(2, R). Further information about the structure of G is obtained; for example, we compute the minimal subgroup of finite index in SL(2, R) containing G.

Original language	English
Title of host publication	Computational Aspects of Discrete Subgroups of Lie Groups
Editors	Alla Detinko, Michael Kapovich, Alex Kontorovich, Peter Sarnak, Richard Schwartz
Publisher	American Mathematical Society
Pages	47-56
Number of pages	10
Volume	783
ISBN (Print)	9781470468040
Publication status	Published - 1 May 2023
Event	Computational Aspects of Discrete Subgroups of Lie Groups - ICERM, Brown University, Providence, United States Duration: 14 Jun 2021 → 18 Jun 2021 https://icerm.brown.edu/topical_workshops/tw-21-cads/

Publication series

Name	Contemporary Mathematics
Publisher	American Mathematical Society
Volume	783
ISSN (Print)	0271-4132
ISSN (Electronic)	1098-3627

Conference

Conference	Computational Aspects of Discrete Subgroups of Lie Groups
Country/Territory	United States
City	Providence
Period	14/06/21 → 18/06/21
Internet address	https://icerm.brown.edu/topical_workshops/tw-21-cads/

Cite this

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title = "Freeness and S-arithmeticity of rational M{\"o}bius groups",

abstract = "We initiate a new, computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) M{\"o}bius subgroups of SL(2, Q). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, R) for a localization R = Z[1b] of Z. We prove that a M{\"o}bius group G ≤ SL(2, R) is not free by showing that it has finite index in SL(2, R). Further information about the structure of G is obtained; for example, we compute the minimal subgroup of finite index in SL(2, R) containing G.",

keywords = "Algorithm, Zariski dense groups, rational M{\"o}bius groups",

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

note = "Funding Information: The third author{\textquoteright}s work has been supported in part by NSF Grant DMS-1720146 and Simons Foundation Grant 852063, which are gratefully acknowledged. Publisher Copyright: {\textcopyright} 2023 American Mathematical Society.; Computational Aspects of Discrete Subgroups of Lie Groups<br/> ; Conference date: 14-06-2021 Through 18-06-2021",

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language = "English",

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series = "Contemporary Mathematics",

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editor = "Alla Detinko and Michael Kapovich and Alex Kontorovich and Peter Sarnak and Richard Schwartz",

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Detinko, AS, Flannery, DL & Hulpke, A 2023, Freeness and S-arithmeticity of rational Möbius groups. in A Detinko, M Kapovich, A Kontorovich, P Sarnak & R Schwartz (eds), Computational Aspects of Discrete Subgroups of Lie Groups. vol. 783, Contemporary Mathematics, vol. 783, American Mathematical Society, pp. 47-56, Computational Aspects of Discrete Subgroups of Lie Groups
, Providence, Rhode Island, United States, 14/06/21.

Freeness and S-arithmeticity of rational Möbius groups. / Detinko, A. S.; Flannery, D. L.; Hulpke, A.

Computational Aspects of Discrete Subgroups of Lie Groups. ed. / Alla Detinko; Michael Kapovich; Alex Kontorovich; Peter Sarnak; Richard Schwartz. Vol. 783 American Mathematical Society, 2023. p. 47-56 (Contemporary Mathematics; Vol. 783).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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T1 - Freeness and S-arithmeticity of rational Möbius groups

AU - Detinko, A. S.

AU - Flannery, D. L.

AU - Hulpke, A.

N1 - Funding Information: The third author’s work has been supported in part by NSF Grant DMS-1720146 and Simons Foundation Grant 852063, which are gratefully acknowledged. Publisher Copyright: © 2023 American Mathematical Society.

PY - 2023/5/1

Y1 - 2023/5/1

N2 - We initiate a new, computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, Q). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, R) for a localization R = Z[1b] of Z. We prove that a Möbius group G ≤ SL(2, R) is not free by showing that it has finite index in SL(2, R). Further information about the structure of G is obtained; for example, we compute the minimal subgroup of finite index in SL(2, R) containing G.

AB - We initiate a new, computational approach to a classical problem: certifying non-freeness of (2-generator, parabolic) Möbius subgroups of SL(2, Q). The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of SL(2, R) for a localization R = Z[1b] of Z. We prove that a Möbius group G ≤ SL(2, R) is not free by showing that it has finite index in SL(2, R). Further information about the structure of G is obtained; for example, we compute the minimal subgroup of finite index in SL(2, R) containing G.

KW - Algorithm

KW - Zariski dense groups

KW - rational Möbius groups

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UR - https://doi.org/10.1090/conm/783

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AN - SCOPUS:85150838044

SN - 9781470468040

VL - 783

T3 - Contemporary Mathematics

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EP - 56

BT - Computational Aspects of Discrete Subgroups of Lie Groups

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A2 - Kapovich, Michael

A2 - Kontorovich, Alex

A2 - Sarnak, Peter

A2 - Schwartz, Richard

PB - American Mathematical Society

T2 - Computational Aspects of Discrete Subgroups of Lie Groups<br/>

Y2 - 14 June 2021 through 18 June 2021

ER -

Freeness and S-arithmeticity of rational Möbius groups

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