## Fricke coordinates

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).

Lemma 2.17   Let be a torsion-free Fuchsian group with compact, an injective homomorphism with discrete image. Then all non-identity elements of are hyperbolic.

If as above is compact of genus (), we can take a set of 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.

Prop 2.18   A Fricke normalized monomorphism with discrete image is determined by the tuple

Theorem 2.19   is a real analytic embedding with real analytic inverse of onto an open domain (where has a complex structure explained later).

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:

We also have that the mapping is a real analytic embedding onto an open domain. However in this case we have that the elements of the group (with discrete image) are either hyperbolic or parabolic (the latter corresponding to loops are around the punctures; they are conjugate in to one of the elements ).

Pablo Ares Gastesi 2005-08-31