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

Mark D. Baker (1) , Daryl Cooper, (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Mathematics Department, University of California

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.

Domaines: Topologie géométrique [math.GT]

Mark D. Baker, Daryl Cooper,. A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds. Journal of topology, 2008, 3, pp.603-642. ⟨10.1112/jtopol/jtn013⟩. ⟨hal-00465985⟩