## Publicacions Matemàtiques

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

### Separation in the BNSR-Invariants of the Pure Braid Groups

#### Abstract

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

**Source**

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

**Dates**

Received: 20 August 2015

Revised: 15 January 2016

First available in Project Euclid: 29 June 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.pm/1498701615

**Digital Object Identifier**

doi:10.5565/PUBLMAT6121702

**Mathematical Reviews number (MathSciNet)**

MR3677865

**Zentralblatt MATH identifier**

06781945

**Subjects**

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Secondary: 57M07: Topological methods in group theory 20F36: Braid groups; Artin groups

**Keywords**

Braid group BNSR-invariant Finiteness properties

#### Citation

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. https://projecteuclid.org/euclid.pm/1498701615