##

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