{"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-property-t-for-operatorname-aut-f-n-and","title":"On property (T) for $\\operatorname{Aut}(F_n)$ and $\\operatorname{SL}_n(\\mathbb{Z})$","arxiv_id":"1812.03456","date":"2018-12-09","proceeding":null,"authors":["Marek Kaluba","Dawid Kielak","Piotr W. Nowak"],"abstract":"We prove that $\\operatorname{Aut}(F_n)$ has Kazhdan's property (T) for every $n \\geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\\geqslant 5$. We also provide explicit lower bounds for the Kazhdan constants of $\\operatorname{SAut}(F_n)$ (with $n \\geqslant 6$) and of $\\operatorname{SL}_n(\\mathbb{Z})$ (with $n \\geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n > 6$.","url_abs":"https://arxiv.org/abs/1812.03456v2","url_pdf":"https://arxiv.org/pdf/1812.03456v2.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-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/kalmarek/1812.03456","is_official":1,"mentioned_in_paper":0,"mentioned_in_github":0,"framework":"none","reach":null},{"paper_slug":"on-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/juliaalgebra/staralgebras.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/kalmarek/2306.12358","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/kalmarek/StarAlgebras.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/kalmarek/groups.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/kalmarek/propertyt.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null},{"paper_slug":"on-property-t-for-operatorname-aut-f-n-and","repo_url":"https://github.com/kalmarek/symbolicwedderburn.jl","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}