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

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 $n\ge 2$, 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 $n\ge 5$) 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 $H\le Aut(P_n(\mathbb{K}))$ such that the Reidemeister number $R(\phi )$ is infinite for every $\phi \in H$.