Abstract
In this paper, we study the relations between the Thurston metric and the hyperbolic metric on a closed surface of genus . We show a rigidity result which says if there is an inequality between the marked length spectra of these two metrics, then they are isotopic. We obtain some inequalities on length comparisons between these metrics. Besides, we show certain distance distortions under conformal graftings, with respect to the metric, the length spectrum metric and Thurston`s asymmetric metrics.
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