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By Baker M., Cooper D.

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If α is a slope and for some n > 0 the immersed slope n · α is an immersed boundary slope, we say that α is a multiple immersed boundary slope or MIBS. Hatcher showed [19] that if M is compact and has boundary a torus, then there are only ﬁnitely many boundary slopes. In [3] it is shown that if M is Seifert ﬁbred, then every immersed boundary slope is also a boundary slope and there are only two boundary slopes; the longitude (rationally null-homologous slope) and the slope of a ﬁbre. The same result holds for MIBS.

We can now apply the virtual amalgam theorem. Thus there are ﬁnite index subgroups Γi ⊂ Γi such that the subgroup Γ ⊂ π1 N generated by Γ1 and Γ2 is their amalgamated free product Γ = Γ1 ∗Γ 0 Γ2 . The hypotheses imply that Γ0 has inﬁnite index in both Γ1 and Γ2 . Furthermore, Γ0 = Γ1 ∩ Γ2 has ﬁnite index in Γ0 . Since Γ0 is torsion-free and non-trivial, it follows that Γ0 is non-trivial. Therefore this is a non-trivial amalgamated free product decomposition. The groups Γ1 , Γ2 are surface groups, and therefore freely indecomposable.

The next result is similar to, but has a stronger conclusion than, a special case of Corollary 5 of Gitik’s paper [16], and also (with a little work) the combination theorem of Bestvina and Feighn [4]. 6 (amalgamating QF subgroups). Let N be a closed hyperbolic 3-manifold. Suppose that Γ1 , Γ2 are two quasi-Fuchsian subgroups of π1 N , each isomorphic to the fundamental group of a closed surface with negative Euler characteristic. Suppose that Γ0 = Γ1 ∩ Γ2 is not trivial, and has inﬁnite index in both Γ1 and in Γ2 .

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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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