{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/freeness-and-s-arithmeticity-of-rational","title":"Freeness and $S$-arithmeticity of rational Möbius groups","arxiv_id":"2203.17201","date":"2022-03-31","proceeding":null,"authors":["A. S. Detinko","D. L. Flannery","A. Hulpke"],"abstract":"We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\\\"{o}bius subgroups of $\\mathrm{SL}(2,\\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\\mathrm{SL}(2, R)$ for a localization $R= \\mathbb{Z}[\\frac{1}{b}]$ of $\\mathbb{Z}$. We prove that a M\\\"{o}bius subgroup $G$ is not free by showing that it has finite index in the relevant $\\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in $\\mathrm{SL}(2,R)$ that contains $G$.","url_abs":"https://arxiv.org/abs/2203.17201v2","url_pdf":"https://arxiv.org/pdf/2203.17201v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"freeness-and-s-arithmeticity-of-rational","repo_url":"https://github.com/hulpke/arithmetic","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}