{"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/presentations-for-cusped-arithmetic","title":"Presentations for cusped arithmetic hyperbolic lattices","arxiv_id":"1709.06691","date":"2017-09-20","proceeding":null,"authors":["Alice Mark","Julien Paupert"],"abstract":"We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice $\\Gamma$, applying a classical result of Macbeath to a suitable $\\Gamma$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups ${\\rm PU}(2,1,\\mathcal{O}_d)$ for $d=1,3,7$ and the quaternion hyperbolic lattice ${\\rm PU}(2,1,\\mathcal{H})$ with entries in the Hurwitz integer ring $\\mathcal{H}$. The implementation of the method for these groups is computer-assisted.","url_abs":"https://arxiv.org/abs/1709.06691v5","url_pdf":"https://arxiv.org/pdf/1709.06691v5.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":"presentations-for-cusped-arithmetic","repo_url":"https://github.com/alice-mark/LatticePresentations","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}