{"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/three-dimensional-maps-and-subgroup-growth","title":"Three-dimensional maps and subgroup growth","arxiv_id":"1712.01418","date":"2017-12-04","proceeding":null,"authors":["Rémi Bottinelli","Laura Ciobanu","Alexander Kolpakov"],"abstract":"In this paper we derive a generating series for the number of cellular complexes known as pavings or three-dimensional maps, on $n$ darts, thus solving an analogue of Tutte's problem in dimension three. The generating series we derive also counts free subgroups of index $n$ in $\\Delta^+ = \\mathbb{Z}_2*\\mathbb{Z}_2*\\mathbb{Z}_2$ via a simple bijection between pavings and finite index subgroups which can be deduced from the action of $\\Delta^+$ on the cosets of a given subgroup. We then show that this generating series is non-holonomic. Furthermore, we provide and study the generating series for isomorphism classes of pavings, which correspond to conjugacy classes of free subgroups of finite index in $\\Delta^+$. Computational experiments performed with software designed by the authors provide some statistics about the topology and combinatorics of pavings on $n\\leq 16$ darts.","url_abs":"https://arxiv.org/abs/1712.01418v7","url_pdf":"https://arxiv.org/pdf/1712.01418v7.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":"three-dimensional-maps-and-subgroup-growth","repo_url":"https://github.com/bottine/nem","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"three-dimensional-maps-and-subgroup-growth","repo_url":"https://github.com/sashakolpakov/monty-3d","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}