Freeness and S-arithmeticity of rational Möbius groups

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Citation (Scopus)


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 languageEnglish
Title of host publicationComputational Aspects of Discrete Subgroups of Lie Groups
EditorsAlla Detinko, Michael Kapovich, Alex Kontorovich, Peter Sarnak, Richard Schwartz
PublisherAmerican Mathematical Society
Number of pages10
ISBN (Print)9781470468040
Publication statusPublished - 1 May 2023
EventComputational Aspects of Discrete Subgroups of Lie Groups
- ICERM, Brown University, Providence, United States
Duration: 14 Jun 202118 Jun 2021

Publication series

NameContemporary Mathematics
PublisherAmerican Mathematical Society
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627


ConferenceComputational Aspects of Discrete Subgroups of Lie Groups
Country/TerritoryUnited States
Internet address


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