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R. Gitik, ‘Ping-pong on negatively curved groups’, J. Algebra 217 (1999) 65–72. 17. J. Hass, J. H. Rubinstein and S. Wang, ‘Boundary slopes of immersed surfaces in 3-manifolds’, J. Differential Geom. 52 (1999) 303–325. 18. J. Hass, S. Wang and Q. Zhou, ‘On finiteness of the number of boundary slopes of immersed surfaces in 3-manifolds’, Proc. Amer. Math. Soc. 130 (2002) 1851–1857 19. A. Hatcher, ‘On the boundary curves of incompressible surfaces’, Pacific J. Math. 99 (1982) 373–377. 20. J. Hempel, ‘The finitely generated intersection property for Kleinian groups’, Knot theory and manifolds, Vancouver, BC, 1983.

2. There are two incompressible, ∂-incompressible, quasi-Fuchsian surfaces S1 , S2 in M with boundary slopes α1 = α2 . The existence of the following cover is perhaps independently interesting. 3. Suppose that M is a compact, orientable 3-manifold with boundary, a torus T , and that the interior of M admits a complete hyperbolic structure of finite volume. Suppose ˜ is finitely ˜ → M such that π1 M that ∂M = T is a torus. Then there is an infinite cover p : M ˜ generated and there are distinct tori T1 , .

Each component of T1 ∩ S˜i is a loop α ˜ i that covers αi . Clearly, α ˜1 , α ˜ 2 generate H1 (T1 ; Q). The result follows from consideration of the algebraic sum of m copies of S˜1 and n copies of S˜2 . 4 (all slopes are MIBS). Suppose that M is a compact, orientable 3-manifold with boundary a torus T , and that the interior of M admits a complete hyperbolic structure of finite volume. Then there is a subgroup of finite index in H1 (T ; Z) such that every non-trivial THE CONVEX APPLICATION THEOREM WITH APPLICATIONS 641 element in this subgroup is an immersed boundary slope for a geometrically finite surface with exactly two boundary components.

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A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds by Baker M., Cooper D.


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