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Additional info for A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds
Deﬁne Ni = Thκ (Mi ); then fi extends to a local isometry fi : Ni → W . Applying the virtual simple gluing theorem to N1 , N2 mapped into W , it follows ˜ i → Mi be the that there are ﬁnite covers pi : Yi → Ni that have a simple gluing Y . Let pi | : M ˜ 1 ⊂ Y1 restriction of the covering pi . We can now apply the convex combination theorem to M ˜ ˜ and M2 ⊂ Y2 and deduce that M has a convex thickening. The next result is similar to, but has a stronger conclusion than, a special case of Corollary 5 of Gitik’s paper , and also (with a little work) the combination theorem of Bestvina and Feighn .
Let N = Core(S + ). In  it was observed that given a rank-2 cusp C of N , every suﬃciently large Dehn-ﬁlling of C can be given a Riemannian metric of negative sectional curvature that agrees with the original metric outside C. Suppose that f : N → M is a local isometry and N + is a Dehn-ﬁlling of N along C. The same ﬁlling done on the corresponding cusp of M then gives a local isometry N + → M + of Dehn-ﬁlled manifolds. If this is done to all the cusps of N then, since N + is convex, this map is π1 -injective.
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