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By Baker M., Cooper D.

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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. A free product of freely indecomposable groups, amalgamated along a non-trivial subgroup, is freely indecomposable.

Then M has a path-isometric embedding onto a spine of a convex hyperbolic n = (m1 + m2 − p)-manifold. Proof. Let Hi ≡ Hm i ⊂ Hn be a totally geodesic subspace with H1 orthogonal to H2 . We can identify the universal cover of P with H1 ∩ H2 in a way that is compatible with identifications of the universal cover of Mi with Hi . Then Yi = T (Mi ; n, κ) is a thickening whose universal cover, Y˜i , is identified with the κ-neighbourhood of Hi . Since the normal bundle of P has trivial holonomy in M1 and M2 , it follows that π1 P preserves both H1 and H2 .

Then there exist a finite cover C˜2 of C2 and a geometrically finite 3-manifold N + = N ∪ C˜2 with a convex thickening. Furthermore, N ∩ C˜2 = C1 , where C1 ⊂ N is identified with a subset of C˜2 using a lift of f |C1 . Proof. Refer to Figure 9. 2. Remark. It is easy to extend this result to the setting where one has immersions of finitely many geometrically finite manifolds, fi : Ni → M , and finitely many rank-2 cusps C2,j ⊂ M , and it is required to glue some of the rank-1 cusps in fi−1 (∪j C2,j ) ⊂ Ni to cyclic covers of C2,j in such a way that if more than one rank-1 cusp is glued onto the same rank-2 cusp, then the rank-1 cusps are glued along parallel curves sufficiently far apart in the boundary of the rank-2 cusps.

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