The Teichmüller space will be obtained using two symplectic reductions. The first one uses a classical non-equivariant momentum map. The second one uses a differential character-valued equivariant momentum map and is associated to the two sheeted hyperboloid adjoint orbit of SL(2,R). This approach leads hence to the study of other spaces that are Teichmüller-type and are associated to the remaining adjoint orbits of SL(2,R).