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Additional info for A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds
W. P. Thurston, Three-dimensional geometry and topology (ed. Silvio Levy; Princeton University Press, Princeton, NJ, 1997). edu.
33. 34. 35. THE CONVEX APPLICATION THEOREM WITH APPLICATIONS A. I. Malcev, ‘On isomorphic matrix representations of inﬁnite groups’, Mat. Sb. 8 (1940) 405–422. B. Maskit, ‘On Klein’s combination theorem. IV’. Trans. Amer. Math. Soc. 336 (1993) 265–294. J. D. Masters, ‘Thick surfaces in hyperbolic 3-manifolds’, Geom. Dedicata 119 (2006) 17–33. K. Matsuzaki and M. Taniguchi, Hyperbolic manifolds and Kleinian groups (Oxford University Press, Oxford, 1998). U. Oertel, ‘Boundaries of π 1 -injective surfaces’ Topology Appl.
Suppose that f : S → M is a continuous map of a closed surface with χ(S) 0 into an irreducible 3-manifold M and suppose that f∗ : π1 S → π1 M is injective. Suppose that f∗ (π1 S) is a separable subgroup of π1 M . Then ˜ → M and an embedding g : S → M ˜ such that p ◦ g is homotopic there exist a ﬁnite cover p : M to f . Proof. By the theorem f∗ (π1 S) is virtually embedded, so there exist a ﬁnite cover p : ˜ → M and an incompressible compact submanifold Y ⊂ M ˜ such that p∗ (π1 Y ) = f∗ (π1 S). M Since π1 Y is a surface group, Y must have non-empty boundary.