Geometry & Topology 7 (2003) 33–90

DOI: 10.2140/gt.2003.7.33

arXiv: math.GT/0107035

Abstract

We characterize which cobounded quasigeodesics in the Teichmüller space T of a closed surface are at bounded distance from a geodesic. More generally, given a cobounded lipschitz path γ in T , we show that γ is a quasigeodesic with finite Hausdorff distance from some geodesic if and only if the canonical hyperbolic plane bundle over γ is a hyperbolic metric space. As an application, for complete hyperbolic 3–manifolds N with finitely generated, freely indecomposable fundamental group and with bounded geometry, we give a new construction of model geometries for the geometrically infinite ends of N, a key step in Minsky’s proof of Thurston’s ending lamination conjecture for such manifolds.

Keywords

Teichmüller space, hyperbolic space, quasigeodesics, ending laminations

Mathematical Subject Classification

Primary: 57M50

Secondary: 32G15

References

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Publication

Received: 15 November 2001
Revised: 6 January 2003
Accepted: 31 2003
Published: 1 February 2003
Proposed: Walter Neumann
Seconded: Benson Farb, David Gabai

Authors