{"id":740,"date":"2025-02-05T18:25:57","date_gmt":"2025-02-05T18:25:57","guid":{"rendered":"https:\/\/as.vanderbilt.edu\/newmath2025\/?p=740"},"modified":"2025-04-17T15:12:25","modified_gmt":"2025-04-17T15:12:25","slug":"topology-and-group-theory-seminar-february-5-2025","status":"publish","type":"post","link":"https:\/\/as.vanderbilt.edu\/math\/2025\/02\/05\/topology-and-group-theory-seminar-february-5-2025\/","title":{"rendered":"Topology and Group Theory Seminar: February 5, 2025"},"content":{"rendered":"<p><strong>Speaker: <\/strong>Itamar Vigdorovich (UCSD)<\/p>\n<p><strong>Title:<\/strong><strong>\u00a0<\/strong><em>Effective mixed identity freeness for higher rank lattices and applications to C*-algebras\u00a0<\/em><\/p>\n<p><strong>Abstract:<\/strong><strong>\u00a0<\/strong>An identity on a group\u00a0<em>G<\/em>\u00a0is a word\u00a0<em>w<\/em>\u00a0that holds throughout the entire group. If\u00a0<em>w<\/em>\u00a0is allowed to include coefficients from\u00a0<em>G<\/em>\u00a0(not just variables), it is called a\u00a0<em>mixed identity<\/em>. We show that a lattice\u00a0<em>\u0393<\/em>\u00a0in\u00a0<em>PSL(n,<\/em><em>\u00a0<\/em><strong><em>R<\/em><\/strong><em>)<\/em><em>\u00a0<\/em>has no non-trivial mixed identities in a quantitative and uniform manner: for any\u00a0<em>r<\/em>, there exists an element\u00a0<em>\u03b3<\/em>\u00a0of linear length in\u00a0<em>r<\/em><em>\u00a0<\/em>that violates all non-trivial mixed words of length at most\u00a0<em>r<\/em>. This has powerful\u00a0<em>C*<\/em>-algebraic applications due to the recent breakthrough of Amrutam, Gao, Kunnawalkam-Elayavalli, and Patchell. Indeed, when combined with the rapid decay property (which is known, for example, for all cocompact lattices in\u00a0<em>SL(3,<\/em><em>\u00a0<\/em><strong><em>R<\/em><\/strong><em>)<\/em>), we deduce that the reduced\u00a0<em>C*<\/em>-algebra of the lattice satisfies strict comparison, stable rank\u00a0<em>1<\/em>,\u00a0<em>K<\/em><em><sub>0<\/sub><\/em>-stability under ultrapowers, uniqueness of the Jiang-Su embedding, and other key properties essential for\u00a0<em>C*<\/em>-classification purposes.<\/p>\n<p><strong>Host:<\/strong><strong>\u00a0<\/strong>Denis Osin<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Speaker: Itamar Vigdorovich (UCSD) Title:\u00a0Effective mixed identity freeness for higher rank lattices and applications to C*-algebras\u00a0 Abstract:\u00a0An identity on a group\u00a0G\u00a0is a word\u00a0w\u00a0that holds throughout the entire group. If\u00a0w\u00a0is allowed to include coefficients from\u00a0G\u00a0(not just variables), it is called a\u00a0mixed identity. We show that a lattice\u00a0\u0393\u00a0in\u00a0PSL(n,\u00a0R)\u00a0has no non-trivial mixed identities in a quantitative and uniform&#8230;<\/p>\n","protected":false},"author":74,"featured_media":736,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":[],"categories":[12],"tags":[],"acf":[],"jetpack_featured_media_url":"https:\/\/as.vanderbilt.edu\/math\/wp-content\/uploads\/sites\/69\/2025\/03\/topology-grouptheory.png","_links":{"self":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/740"}],"collection":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/users\/74"}],"replies":[{"embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/comments?post=740"}],"version-history":[{"count":1,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/740\/revisions"}],"predecessor-version":[{"id":741,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/740\/revisions\/741"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/media\/736"}],"wp:attachment":[{"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/media?parent=740"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/categories?post=740"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/tags?post=740"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}