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

We prove the convex combination theorem for hyperbolic n-manifolds. Applications are given both in high dimensions and in three dimensions. One consequence is that given two geometrically finite subgroups of a discrete group of isometries of hyperbolic n-space, satisfying a natural condition on their parabolic subgroups and intersection with a separable subgroup, there are finite index subgroups which generate a subgroup that is an amalgamated free product. Constructions of infinite volume hyperbolic n-manifolds are described by gluing lower H dimensional manifolds. It is shown that every slope on a cusp of a hyperbolic 3-manifold is a multiple immersed boundary slope. If the fundamental group of a hyperbolic 3-manifold contains a maximal surface group not carried by an embedded surface, then it contains a freely indecomposable group with second Betti number at least 2.