Comparison between Teichmuller and Lipschitz metrics |
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Authors: | Choi Young-Eun; Rafi Kasra |
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Institution: | Penn State Altoona 3000 Ivyside Park Altoona, PA 16601 USA choiye{at}psu.edu
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Abstract: | We study the Lipschitz metric on a Teichmüller space (definedby Thurston) and compare it with the Teichmüller metric.We show that in the thin part of the Teichmüller spacethe Lipschitz metric is approximated up to a bounded additivedistortion by the sup-metric on a product of lower-dimensionalspaces (similar to the Teichmüller metric as shown by Minsky).In the thick part, we show that the two metrics are equal upto a bounded additive error. However, these metrics are notcomparable in general; we construct a sequence of pairs of pointsin the Teichmüller space, with distances that approachzero in the Lipschitz metric while they approach infinity inthe Teichmüller metric. |
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