Automorphisms of braid groups on orientable surfaces

Abstract

In this paper, we compute the automorphism groups Aut(Pn(Σ)) and Aut(Bn(Σ)) of braid groups Pn(Σ) and Bn(Σ) on every orientable surface Σ, which are isomorphic to group extensions of the extended mapping class group M∗n(Σ) by the transvection subgroup except for a few cases. We also prove that Pn(Σ) is always a characteristic subgroup of Bn(Σ), unless Σ is a twice-punctured sphere and n = 2. (c) World Scientific Publishing Company