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**Additional info for A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds**

**Sample text**

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 [16], and also (with a little work) the combination theorem of Bestvina and Feighn [4].

Let N = Core(S + ). In [8] 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.

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. Lecture Notes in Mathematics 1144 (Springer, Berlin, 1985) 18–24. 21. E. Kang and J. H. Rubinstein, ‘Ideal triangulations of 3-manifolds I: spun normal surface theory’, Geom. Topol. Monogr. 7 (2004) 235–265. 22. D. D. ’ Bull. London Math. Soc.