{"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/groups-of-order-64-and-non-homeomorphic","title":"Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants","arxiv_id":"2412.08260","date":"2024-12-11","proceeding":null,"authors":["Francesco Polizzi","Pietro Sabatino"],"abstract":"Let $\\Sigma_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands $\\mathsf{P}_2(\\Sigma_b)$ which do not factor through $\\pi_1(\\Sigma_b \\times \\Sigma_b)$. Building on our previous work on some special systems of generators on finite groups that we called \\emph{diagonal double Kodaira structures}, we prove that, if $G$ has not order $32$, then $|G| \\geq 64$, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group.","url_abs":"https://arxiv.org/abs/2412.08260v2","url_pdf":"https://arxiv.org/pdf/2412.08260v2.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":"groups-of-order-64-and-non-homeomorphic","repo_url":"https://github.com/pietros16bit/gap4ddks","is_official":1,"mentioned_in_paper":1,"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}