We now want to give some coordinates to , for a compact surface, possible with punctures. We start with a lemma that says that all non-trivial elements of a Fuchsian group uniformizing a compact surface are of the same type, namely hyperbolic transformations (see subsection 1.2).
standard loops, , , which generate the fundamental group of (which is isomorphic to ) and satisfy the relation (here denotes the commutator of two elements). Denote by the same letters the corresponding set of generators of . We can normalized these generators so that has attractive fixed point at and repelling fixed point at 0 while has attractive fixed point at . If is an injective group homomorphism then we require that and are also normalized in this form. Such homomorphism is called a normalized Fricke homomorphism. Consider now the sequence of generators of given by . Each of these transformations can be represented by a matrix with real entries, satisfying and if . Moreover, if the image of is discrete then these transformations are hyperbolic.
In the case of a surface with punctures we have a similar result, where the image of lies in . The group is generated by elements, say , and , satisfying . The elements and are hyperbolic; we require that for any matrix representing any of these elements, we have and if . The elements are parabolic; our requirements in this case are that the matrix representing satisfies . The a Fricke normalized homomorphism is determined uniquely by the following tuple of real numbers:
Pablo Ares Gastesi 2005-08-31