Finding disjoint surfaces in 3-manifolds

Let M be a compact connected orientable 3-manifold, with non-empty boundary that contains no 2-spheres. We investigate the existence of two properly embedded disjoint surfaces S_1 and S_2 such that M - (S_1 \cup S_2) is connected. We show that there exist two such surfaces if and only if M is neither a Z_2 homology solid torus nor a Z_2 homology cobordism between two tori. In particular, the exterior of a link with at least 3 components always contain two such surfaces. The proof mainly uses techniques from the theory of groups, both discrete and profinite.