{"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/noncommutative-real-algebraic-geometry-of","title":"Noncommutative real algebraic geometry of Kazhdan's property (T)","arxiv_id":"1312.5431","date":"2013-12-19","proceeding":null,"authors":["Narutaka Ozawa"],"abstract":"It is well-known that a finitely generated group $\\Gamma$ has Kazhdan's property (T) if and only if the Laplacian element $\\Delta$ in ${\\mathbb R}[\\Gamma]$ has a spectral gap. In this paper, we prove that this phenomenon is witnessed in ${\\mathbb R}[\\Gamma]$. Namely, $\\Gamma$ has property (T) if and only if there are a constant $\\kappa>0$ and a finite sequence $\\xi_1,...,\\xi_n$ in ${\\mathbb R}[\\Gamma]$ such that $\\Delta^2-\\kappa\\Delta = \\sum_i \\xi_i^*\\xi_i$. This result suggests the possibility of finding new examples of property (T) groups by solving equations in ${\\mathbb R}[\\Gamma]$, possibly with an assist of computers.","url_abs":"http://arxiv.org/abs/1312.5431v3","url_pdf":"http://arxiv.org/pdf/1312.5431v3.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":"noncommutative-real-algebraic-geometry-of","repo_url":"https://github.com/kalmarek/propertyt.jl","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}