{"id":744,"date":"2025-01-22T18:35:19","date_gmt":"2025-01-22T18:35:19","guid":{"rendered":"https:\/\/as.vanderbilt.edu\/newmath2025\/?p=744"},"modified":"2025-04-17T15:12:25","modified_gmt":"2025-04-17T15:12:25","slug":"topology-and-group-theory-seminar-january-22-2025","status":"publish","type":"post","link":"https:\/\/as.vanderbilt.edu\/math\/2025\/01\/22\/topology-and-group-theory-seminar-january-22-2025\/","title":{"rendered":"Topology and Group Theory Seminar: January 22, 2025"},"content":{"rendered":"<p><strong>Speaker: <\/strong>Mike Mihalik\u00a0(Vanderbilt)<\/p>\n<p><strong>Title:<\/strong><strong>\u00a0<\/strong><em>Stallings\u2019 group is simply connected at infinity<\/em><\/p>\n<p><strong>Abstract:<\/strong><strong>\u00a0<\/strong>For\u00a0<em>n \u2265 2<\/em>, let\u00a0<em>B<\/em><em><sub>n<\/sub><\/em>\u00a0denote the kernel of the homomorphism from the direct product of\u00a0<em>n<\/em>-copies of the free group on two generators to the group of integers which sends all generators to the generator 1. The groups\u00a0<em>B<\/em><em><sub>n<\/sub><\/em>\u00a0are called the Bieri-Stallings groups and\u00a0<em>B<\/em><em><sub>n<\/sub><\/em>\u00a0is of type\u00a0<em>F<\/em><em><sub>n-1<\/sub><\/em>\u00a0but not\u00a0<em>F<\/em><em><sub>n<\/sub><\/em>. Classical results can be used to show that\u00a0<em>B<\/em><em><sub>n<\/sub><\/em>\u00a0is\u00a0<em>(n-3)-<\/em>connected at infinity for\u00a0<em>n \u2265 3<\/em>. Since\u00a0<em>B<\/em><em><sub>n<\/sub><\/em>\u00a0is not type\u00a0<em>F<\/em><em><sub>n<\/sub><\/em>\u00a0it is not\u00a0<em>(n-1)-<\/em>connected at infinity. Stallings\u2019 proved that\u00a0<em>B<\/em><em><sub>2<\/sub><\/em>\u00a0is finitely generated but not finitely presented. We conjecture that for\u00a0<em>n \u2265 2<\/em>,\u00a0<em>B<\/em><em><sub>n<\/sub><\/em>\u00a0is\u00a0<em>(n-2)<\/em>-connected at infinity. For\u00a0<em>n = 2<\/em>, this would mean that\u00a0<em>B<\/em><em><sub>2<\/sub><\/em>\u00a0is\u00a0<em>1-<\/em>ended and for\u00a0<em>n = 3<\/em>\u00a0that\u00a0<em>B<\/em><em><sub>3<\/sub><\/em><em><sub>\u00a0<\/sub><\/em>(typically called Stallings\u2019 group) is simply connected at infinity. We verify the conjecture for\u00a0<em>n = 2<\/em>\u00a0and\u00a0<em>n = 3<\/em>. Our main result is the case\u00a0<em>n = 3<\/em>: Stalling\u2019s group is simply connected at infinity.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Speaker: Mike Mihalik\u00a0(Vanderbilt) Title:\u00a0Stallings\u2019 group is simply connected at infinity Abstract:\u00a0For\u00a0n \u2265 2, let\u00a0Bn\u00a0denote the kernel of the homomorphism from the direct product of\u00a0n-copies of the free group on two generators to the group of integers which sends all generators to the generator 1. The groups\u00a0Bn\u00a0are called the Bieri-Stallings groups and\u00a0Bn\u00a0is of type\u00a0Fn-1\u00a0but not\u00a0Fn. Classical&#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\/744"}],"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=744"}],"version-history":[{"count":1,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/744\/revisions"}],"predecessor-version":[{"id":745,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/posts\/744\/revisions\/745"}],"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=744"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/categories?post=744"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/as.vanderbilt.edu\/math\/wp-json\/wp\/v2\/tags?post=744"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}