Automorphisms of braid groups on orientable surfaces

Abstract

In this paper we compute the automorphism groups $\mathrm{Aut}({\mathbf{P}}_n(\Sigma ))$ and $\mathrm{Aut}({\mathbf{B}}_n(\Sigma ))$ of braid groups ${\mathbf{P}}_n(\Sigma )$ and ${\mathbf{B}}_n(\Sigma )$ on every orientable surface $\Sigma$, which are isomorphic to group extensions of the extended mapping class group ${\mathcal{M}}_n^{\ast }(\Sigma )$ by the transvection subgroup except for a few cases. We also prove that ${\mathbf{P}}_n(\Sigma )$ is always a characteristic subgroup of ${\mathbf{B}}_n(\Sigma )$ unless $\Sigma$ is a twice-punctured sphere and $n=2$.

27 pages

References in corpus (1)

• Commensurators of surface braid groups

Cited by in corpus (3)

• The R${}_{\infty }$-property for braid groups over orientable surfaces
• Isotopy and homeomorphism of closed surface braids
• Lower central series, surface braid groups, surjections and permutations