A Combination Theorem for Convex Hyperbolic Manifolds, with Applications to Surfaces in 3-Manifolds

Abstract

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.

43 pages, 10 Postscript figures. Minor changes to 2.4, 2.5, 8.1, 8.5. Citations added and corrected

References in corpus (2)

• Combination of convergence groups
• Some surface subgroups survive surgery

Cited by in corpus (10)

• Effective bilipschitz bounds on drilling and filling
• Ubiquitous quasi-Fuchsian surfaces in cusped hyperbolic 3-manifolds
• Virtual Amalgamation of Relatively Quasiconvex Subgroups
• Deforming convex projective manifolds
• A combination theorem for Anosov subgroups
• Finite-volume Hyperbolic 3-Manifolds contain immersed Quasi-Fuchsian surfaces
• Klein-Maskit combination theorem for Anosov subgroups: Free products
• Combination of fully quasiconvex subgroups and its applications
• Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups
• A quadratic bound on the number of boundary slopes of essential surfaces with bounded genus