{"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/inducing-spectral-gaps-for-the-cohomological","title":"Inducing spectral gaps for the cohomological Laplacians of $\\operatorname{SL}_n(\\mathbb{Z})$ and $\\operatorname{SAut}(F_n)$","arxiv_id":"2404.10287","date":"2024-04-16","proceeding":null,"authors":["Piotr Mizerka"],"abstract":"The technique of inducing spectral gaps for cohomological Laplacians in degree zero was used by Kaluba, Kielak and Nowak to prove property (T) for $\\operatorname{SAut}(F_n)$ and $\\operatorname{SL}_n(\\mathbb{Z})$. In this paper, we adapt this technique to Laplacians in degree one. This allows to provide a lower bound for the cohomological Laplacian in degree one for $\\operatorname{SL}_n(\\mathbb{Z})$ for every unitary representation. In particular, one gets in that way an alternative proof of property (T) for $\\operatorname{SL}_n(\\mathbb{Z})$ whenever $n\\geq 3$.","url_abs":"https://arxiv.org/abs/2404.10287v3","url_pdf":"https://arxiv.org/pdf/2404.10287v3.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":"inducing-spectral-gaps-for-the-cohomological","repo_url":"https://github.com/piotrmizerka/lowcohomologysos","is_official":1,"mentioned_in_paper":1,"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}