Surjective homomorphisms between surface braid groups

Let $P{B}_n(S_{g,p})$ be the pure braid group of a genus $g>1$ surface with $p$ punctures. In this paper we prove that any surjective homomorphism $P{B}_n(S_{g,p})\to P{B}_m(S_{g,p})$ factors through one of the forgetful homomorphisms. We then compute the automorphism group of $P{B}_n(S_{g,p})$, extending Irmak, Ivanov and McCarthy’s result to the punctured case. Surprisingly, in contrast to the $n=1$ case, any automorphism of $P{B}_n(S_{g,p})$, $n>1$ is geometric.