Open Access
2017 Separation in the BNSR-Invariants of the Pure Braid Groups
Matthew C. B. Zaremsky
Publ. Mat. 61(2): 337-362 (2017). DOI: 10.5565/PUBLMAT6121702


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 $\mathrm{F}_{m-3}$ but not $\mathrm{F}_{m-2}$, for all $3\le m\le n$.


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Matthew C. B. Zaremsky. "Separation in the BNSR-Invariants of the Pure Braid Groups." Publ. Mat. 61 (2) 337 - 362, 2017.


Received: 20 August 2015; Revised: 15 January 2016; Published: 2017
First available in Project Euclid: 29 June 2017

zbMATH: 06781945
MathSciNet: MR3677865
Digital Object Identifier: 10.5565/PUBLMAT6121702

Primary: 20F65
Secondary: 20F36 , 57M07

Keywords: BNSR-invariant , Braid group , finiteness properties

Rights: Copyright © 2017 Universitat Autònoma de Barcelona, Departament de Matemàtiques

Vol.61 • No. 2 • 2017
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