  We prove the convex combination theorem for hyperbolic n-manifolds.
Applications are given both in high dimensions and in 3 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, 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 dimensional manifolds. It
is shown that every slope on a cusp of a hyperbolic 3-manifold is a multiple
immersed boundary slope. If a 3-manifold contains a maximal surface group not
carried by an embedded surface then it contains the fundamental group of a book
of I-bundles with more than two pages.
