{"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/on-the-arithmetic-dimension-of-triangle","title":"On the arithmetic dimension of triangle groups","arxiv_id":"1510.04637","date":"2015-10-15","proceeding":null,"authors":["Steve Nugent","John Voight"],"abstract":"Let $\\Delta=\\Delta(a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $\\pi/a,\\pi/b,\\pi/c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $\\Delta$ to be the number of split real places of the quaternion algebra generated by $\\Delta$ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $(a,b,c)$ with arithmetic dimension $1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.","url_abs":"https://arxiv.org/abs/1510.04637v2","url_pdf":"https://arxiv.org/pdf/1510.04637v2.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":"on-the-arithmetic-dimension-of-triangle","repo_url":"https://github.com/stevenugent/Find-R-Arithmetic-Triples","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"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}