• DOI:10.5565/PUBLMAT6121702
• Corpus ID: 118492166

Separation in the BNSR-invariants of the pure braid groups

@article{Zaremsky2015SeparationIT,
  title={Separation in the BNSR-invariants of the pure braid groups},
  author={Matthew C. B. Zaremsky},
  journal={arXiv: Group Theory},
  year={2015}
}

• Matthew C. B. Zaremsky
• Published 30 July 2015
• Mathematics
• arXiv: Group Theory

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 $3\le m\le n$.