Consider the Fricke homeomorphism (given by Fricke coordinates on Teichmüller space) and let be a lift of .
This theorem can be used to give a complex structure to the Teichmüller space . For that we need some general facts on mappings between complex and real manifolds, and when such mappings induce complex structures (on the real manifold). Let be a submersion from a complex Banach manifold onto a real manifold . We say that induces a well-defined almost complex structure on if for all the space inherits via the differential a unique complex structure, independent of the choice of .
We can apply the above result to the mapping , define from the unit ball of to , where is the Fricke mapping. To check that induces a well-defined almost complex structure one needs to use the right-translation mappings, , from the unit ball of to itself, defined by
The Fricke homeomorphism gives then a complex structure to Teichmüller space . The mapping becomes biholomorphic and the projection is then a holomorphic submersion.
The discontinuous action of the modular group on Teichmüller space gives us some information about the moduli space of surfaces.
Pablo Ares Gastesi 2005-08-31