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

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