Automorphisms of braid groups on closed surfaces which are not S2, T2, P2 or the Klein bottle

Abstract

Consider a surface braid group of n strings as a subgroup of the isotopy group of homeomorphisms of the surface permuting n fixed distinguished points. Each automorphism of the surface braid group (respectively, of the special surface braid group) is shown to be a conjugate action on the braid group (respectively, on the special braid group) induced by a homeomorphism of the underlying surface if the closed surface, either orientable or non-orientable, is of negative Euler characteristic. In other words, the group of automorphisms of such a surface braid group is isomorphic to the extended mapping class group of the surface with n punctures, while the outer automorphism group of the surface braid group is isomorphic to the extended mapping class group of the closed surface itself.