Geometric subgroups of surface braid groups

Let M be a surface, let N be a subsurface, and let n less than or equal to m he two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group BnN with n strings on N to the braid group BmM with m,strings on M. We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of pi(1)N in pi(1)M. Then we calculate the commensurator, the normalizer and the centralizer of BnN in BmM for large surface braid groups.