Separability of embedded surfaces in 3-manifolds

This is joint work with Dani Wise. Let S be an immersed incompressible surface in a 3-manifold M. Denote by M' the universal cover of M. Scott proved that the group pi_1S is separable in pi_1M iff any compact neighborhood of S in pi_1S\M' embeds in some finite cover of M. Rubinstein and Wang found an immersed surface which does not lift to an embedding in a finite cover, hence violates this condition. We prove that this is the only obstruction, i.e. that if S is already embedded, then pi_1S is separable.