Separation in the BNSR-invariants of the pure braid groups

TL;DR

This paper investigates the structure of BNSR-invariants of pure braid groups, revealing a layered hierarchy of subgroup finiteness properties and providing explicit character classes for each level.

Contribution

It proves the proper inclusion relations among BNSR-invariants for pure braid groups and explicitly characterizes the differences between successive invariants.

Findings

Proper inclusion of $ ext{Sigma}^{m-2}(P_n)$ in $ ext{Sigma}^{m-3}(P_n)$ for $3 \\le m \\le n$.

Equality of $ ext{Sigma}^ ext{infty}(P_n)$ and $ ext{Sigma}^{n-2}(P_n)$.

Explicit character classes distinguishing different BNSR-invariants.

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