Abstract
AbstractWe study the Lipschitz metric on a Teichmuller space (defined by Thurston) and compare it with the Teichmuller metric. We show that in the thin part of the Teichmuller 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 Teichmuller 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 Teichmuller space, with distances that approach zero in the Lipschitz metric while they approach infinity in the Teichmuller metric.
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