Automorphisms of braid groups on orientable surfaces

TL;DR

This paper determines the automorphism groups of braid groups on all orientable surfaces, revealing their structure as extensions of the extended mapping class group, with specific exceptions, and establishes characteristic subgroup relations.

Contribution

It provides a comprehensive computation of automorphism groups of surface braid groups, extending previous results to all orientable surfaces and identifying key subgroup properties.

Findings

Automorphism groups are isomorphic to extensions of the extended mapping class group.

$ extbf{P}_n( ext{Surface})$ is a characteristic subgroup except in a specific case.

Explicit descriptions of automorphism groups for all orientable surfaces.

Abstract

In this paper we compute the automorphism groups $\operatorname{Aut}(\mathbf{P}_n(\Sigma))$ and $\operatorname{Aut}(\mathbf{B}_n(\Sigma))$ 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 $\mathcal{M}^*_n(\Sigma)$ 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 $n=2$.