Extremal Quasiconformal Mappings Compatible 40 with Fuchsian Groups

For every torsion free Fuchsian group . with Poincaré's .-operator norm ||.. ||= 1, it is proved that there exists an extremal Beltrami differential of . which is also extremal under its own boundary correspondence. It is also proved that the imbedding of the Teichmüller space T(T) into the universal Teichmüller space T is not a global isometry unless I is an elementary group.