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 homomor phism from the braid group BnN with n strings on N to the braid group BmM w ith 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 lar ge surface braid groups.