{"paper":{"title":"Separation in the BNSR-invariants of the pure braid groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Matthew C. B. Zaremsky","submitted_at":"2015-07-30T17:51:24Z","abstract_excerpt":"We inspect the BNSR-invariants $\\Sigma^m(P_n)$ of the pure braid groups $P_n$, using Morse theory. The BNS-invariants $\\Sigma^1(P_n)$ were previously computed by Koban, McCammond and Meier. We prove that for any $3\\le m\\le n$, the inclusion $\\Sigma^{m-2}(P_n)\\subseteq \\Sigma^{m-3}(P_n)$ is proper, but $\\Sigma^\\infty(P_n)=\\Sigma^{n-2}(P_n)$. We write down explicit character classes in each relevant $\\Sigma^{m-3}(P_n)\\setminus \\Sigma^{m-2}(P_n)$. In particular we get examples of normal subgroups $N\\le P_n$ with $P_n/N\\cong\\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$, for all "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.08597","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}