Stable Teichmüller quasigeodesics and ending laminations

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

References

Forward citations

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

Volume 7, issue 1 (2003)

Lee Mosher