Let M be a hyperbolic
3-manifold admitting a homotopy equivalence to a compact surface Σ, where the
cusps of M correspond exactly to the boundary components of Σ. We construct
a nested system of bands in M, where each band is homeomorphic to a
subsurface of Σ times an interval. This band system is shown to have various
geometrical properties, notably that the boundary of any Margulis tube is
mostly contained in the union of the bands. As a consequence, one can deduce
the result (conjectured by McMullen and proven by Brock, Canary and
Minsky) that the thick part of the convex core of M has at most polynomial
growth. Moreover the degree is at most minus the Euler characteristic of Σ.
Other applications of this construction to the curve complex of Σ will be
discussed elsewhere. The complex is related to the block decomposition of M
described by Minsky, in his work towards Thurston’s Ending Lamination
Conjecture.
Keywords
3-manifolds, hyperbolic geometry
Mathematical Subject Classification 2000
Primary: 57M50
Milestones
Received: 16 January 2006
Accepted: 19 July 2007
Published: 1 September 2007
Authors
Brian H. Bowditch
Mathematics Institute
University of Warwick
Coventry, CV4 7AL
Great Britain
