Kleinian Groups and Hyperbolic 3-Manifolds
Proceedings of the Warwick Workshop, September 11–14, 2001
Edited by Y. Komori
Edited by V. Markovic
Edited by C. Series
Publisher: Cambridge University Press
Print Publication Year: 2003
Online Publication Date:September 2009
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511542817.006
We study the relationship between the geometry and the topology of the ends of a hyperbolic 3-manifold M whose fundamental group is not a free group. We prove that a compressible geometrically infinite end of M is tame if there is a Masur domain lamination which is not realized by a pleated surface. It is due to Canary that, in the absence of rank-1-cusps, this condition is also necessary.
Marden [Mar74] proved that every geometrically finite hyperbolic 3-manifold is tame, i.e. homeomorphic to the interior of a compact manifold, and he conjectured that this holds for any hyperbolic 3-manifold M with finitely generated fundamental group. By a theorem of Scott [Sco73b], M contains a core, a compact submanifold C such that the inclusion of C into M is a homotopy equivalence. Moreover, any two cores are homeomorphic by a homeomorphism in the correct homotopy class [MMS85].
So the discussion of the tameness of M boils down to a discussion of the ends of M. The ends are in a bijective correspondence with the boundary components of the compact core C. An end E is said to be tame if it has a neighborhood homeomorphic to the product of the corresponding boundary component ∂E of C with the half-line. Hence M is tame if its ends are.
Since M is aspherical, either π1(M) = 1 or every boundary component of C is a closed surface of genus at least 1.