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Stable Teichmüller quasigeodesics and ending laminations

Lee Mosher

Geometry & Topology 7 (2003) 33–90

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 2000
Primary: 57M50
Secondary: 32G15
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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
Lee Mosher
Department of Mathematics and Computer Science
Rutgers University
Newark
New Jersey 07102
USA