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Journal of Topology Advance Access originally published online on June 13, 2008
Journal of Topology 2008 1(3):603-642; doi:10.1112/jtopol/jtn013
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© 2008 London Mathematical Society

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

Mark Baker

Mathematics Department, Université de Rennes 1 France
mark.baker@univ-rennes1.fr

Daryl Cooper

Mathematics Department, University of California, Santa Barbara Santa Barbara, CA 93106 USA
cooper@math.ucsb.edu.


  Abstract

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.

Received March 12, 2007.

2000 Mathematics Subject Classification 57M50 (primary), 30F40 (secondary)

This work was partially supported by NSF grants DMS0104039 and DMS0405963.


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