Surjective homomorphisms between surface braid groups

Let PBn(Sg,p) be the pure braid group of a genus g > 1 surface with p punctures. In this paper we prove that any surjective homomorphism PBn(Sg,p) → PBm(Sg,p) factors through one of the forgetful homomorphisms. We then compute the automorphism group of PBm(Sg,p), which gives a simpler proof of Irmak–Ivanov–McCarthy [IIM03, Theorem 1]. Surprisingly, in contrast to the n = 1 case, any automorphism of PBn(Sg,p) for n > 1 is geometric.