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Journal of the London Mathematical Society Advance Access originally published online on November 20, 2007
Journal of the London Mathematical Society 2007 76(3):739-756; doi:10.1112/jlms/jdm052
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© 2007 London Mathematical Society

Comparison between Teichmüller and Lipschitz metrics

Young-Eun Choi

Penn State Altoona
3000 Ivyside Park
Altoona, PA 16601
USA
choiye{at}psu.edu

Kasra Rafi

Department of Mathematics
196 Auditorium Road, U-3009
University of Connecticut
Storrs, CT 06269-3009
USA

We study the Lipschitz metric on a Teichmüller space (defined by Thurston) and compare it with the Teichmüller metric. We show that in the thin part of the Teichmüller space the Lipschitz metric is approximated up to a bounded additive distortion by the sup-metric on a product of lower-dimensional spaces (similar to the Teichmüller metric as shown by Minsky). In the thick part, we show that the two metrics are equal up to a bounded additive error. However, these metrics are not comparable in general; we construct a sequence of pairs of points in the Teichmüller space, with distances that approach zero in the Lipschitz metric while they approach infinity in the Teichmüller metric.

www.math.uconn.edu/~rafi

2000 Mathematics Subject Classification 30F60 (primary), 32G15, 57M50 (secondary).

Received October 27, 2005; revised December 15, 2006; published online November 20, 2007.


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