The BNS invariants of the braid groups and pure braid groups of some surfaces

TL;DR

This paper explicitly computes the Bieri-Neumann-Strebel invariants for full and pure braid groups on various surfaces, revealing their geometric structure and implications for automorphism groups and Reidemeister numbers.

Contribution

It provides explicit descriptions of the BNS invariants for braid groups on spheres, projective planes, tori, and Klein bottles, including their geometric and algebraic properties.

Findings

$ ext{Sigma}^1$ invariants are finite unions of circles or finite sets.

The invariants are invariant under certain permutations of coordinates.

Existence of subgroups with infinite Reidemeister number for automorphisms.

Abstract

We compute and explicitly describe the Bieri-Neumann-Strebel invariants $\Sigma^1$ for the full and pure braid groups of the sphere $\mathbb{S}^2$, the real projective plane $\mathbb{R}P^2$ and specially the torus $\mathbb{T}$ and the Klein bottle $\mathbb{K}$. In order to do this for $M=\mathbb T$ or $M=\mathbb K$, and $n \geq 2$, we use the $n^{th}$-configuration space of $M$ to show that the action by homeomorphisms of the group $Out(P_n(M))$ on the character sphere $S(P_n(M))$ contains certain permutation of coordinates, under which $\Sigma^1(P_n(\mathbb T))^c$ and $\Sigma^1(P_n(\mathbb K))^c$ are invariant. Furthermore, $\Sigma^1(P_n(\mathbb T))^c$ and $\Sigma^1(P_n(\mathbb{S}^2))^c$ (the latter with $n \geq 5$) are finite unions of pairwise disjoint circles, and $\Sigma^1(P_n(\mathbb K))^c$ is finite. This last fact implies that there is a normal finite index subgroup $H \leq Aut(P_n(\mathbb K))$ such that the Reidemeister number $R(\varphi)$ is infinite for every $\varphi \in H$.