Thurston's metric on Teichm\"uller space and isomorphisms between Fuchsian groups

The aim of this paper is to relate Thurston’s metric on Teichm"uller space to several ideas initiated by T. Sorvali on isomorphisms between Fuchsian groups. In particular, this will give a new formula for Thurston’s asymmetric metric for surfaces with punctures. We also update some results of Sorvali on boundary isomorphisms of Fuchsian groups.