Thurston's metric on Teichmüller space and the translation lengths of mapping classes

Abstract

We show that the Teichmüller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichmüller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances for mapping classes for their actions on Teichmüller spaces equipped with their arc metrics.