Finding disjoint surfaces in 3-manifolds

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no two-spheres. We investigate the existence of two properly embedded disjoint surfaces S1 and S2 such that M - (S1 ∪ S2) is connected. We show that there exist two such surfaces if and only if M is neither a Z2 homology solid torus nor a ℤ2 homology cobordism between two tori. In particular, the exterior of a link with at least three components always contains two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite. © 2013 Springer Science+Business Media Dordrecht.