Comparison between Teichmüller and Lipschitz metrics

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.