  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.
