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Separation in the BNSR-Invariants of the Pure Braid Groups

Matthew C. B. Zaremsky

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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 $3\le m\le n$.

Article information

Publ. Mat., Volume 61, Number 2 (2017), 337-362.

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 57M07: Topological methods in group theory 20F36: Braid groups; Artin groups

Braid group BNSR-invariant Finiteness properties


Zaremsky, Matthew C. B. Separation in the BNSR-Invariants of the Pure Braid Groups. Publ. Mat. 61 (2017), no. 2, 337--362. doi:10.5565/PUBLMAT6121702.

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