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