Fricke coordinates

We now want to give some coordinates to $T(X)$, for $X$ 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 $G$ be a torsion-free Fuchsian group with $X= \mathbb{H}/G$ compact, $E:G\to\mathrm{PSL}(2,\mathbb{R})$ an injective homomorphism with discrete image. Then all non-identity elements of $E(G)$ are hyperbolic.

If $X$ as above is compact of genus $g$ ($\geq 2$), we can take a set of standard loops , $A_1,\ldots,A_g,$, $B_1,\ldots,B_g$, which generate the fundamental group of $X$ (which is isomorphic to $G$) and satisfy the relation $\prod_{j=1}^g [A_j,B_j] = Id$ (here $[A,B] = ABA^{-1}B^{-1}$ denotes the commutator of two elements). Denote by the same letters the corresponding set of generators of $G$. We can normalized these generators so that $B_g$ has attractive fixed point at $\infty$ and repelling fixed point at 0 while $A_g$ has attractive fixed point at $1$. If $E:G\to\mathrm{PSL}(2,\mathbb{R})$ is an injective group homomorphism then we require that $E(B_g)$ and $E(A_g)$ are also normalized in this form. Such homomorphism is called a normalized Fricke homomorphism. Consider now the sequence of generators of $E(G)$ given by $E(A_1), E(B_1),\ldots,E(A_g),E(B_g)$. Each of these transformations can be represented by a matrix with real entries, $(\begin{smallmatrix}a_i & b_i c_i & d_i\end{smallmatrix})$ satisfying $a_i \geq 0$ and $b_i > 0$ if $a_i = 0$. Moreover, if the image of $E$ is discrete then these transformations are hyperbolic.

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

$$(a_1,b_1,c_1,\ldots,a_{2g-2},b_{2g-2},c_{2g-2}) \in {\mathbb{R}}^{6g-6}.$$

Theorem 2.19   $F:T(X) \to {\mathbb{R}}^{6g-6}$ is a real analytic embedding with real analytic inverse of $T(X)$ onto an open domain (where $T(X)$ has a complex structure explained later).

In the case of a surface with punctures we have a similar result, where the image of $F$ lies in ${\mathbb{R}}^{6g-6+2n}$. The group $G$ is generated by $2g+n$ elements, say $A_1,\ldots,A_g$, $B_1,\ldots,B_g$ and $C_1,\ldots,C_n$, satisfying $(\prod_{j=1}^g [A_j,B_j])C_1\cdots C_n = Id$. The elements $A_j$ and $B_j$ are hyperbolic; we require that for any matrix $(\begin{smallmatrix}a_i & b_i c_i & d_i\end{smallmatrix})$ representing any of these elements, we have $a_i \geq 0$ and $b_i > 0$ if $a_i = 0$. The elements $C_i$ are parabolic; our requirements in this case are that the matrix $(\begin{smallmatrix}a'_i & b'_i c'_i &d'_i\end{smallmatrix})$ representing $C_i$ satisfies $a'_i + d'_i = 2$. The a Fricke normalized homomorphism is determined uniquely by the following tuple of real numbers:

$$(a_1,b_1,c_1,\ldots,a_{2g-2},b_{2g-2},c_{2g-2},a'_1,c'_1,\ldots,a'_n,c'_n).$$

We also have that the mapping $F:T(X) \to {\mathbb{R}}^{6g-6+2n}$ is a real analytic embedding onto an open domain. However in this case we have that the elements of the group $E(G)$ (with discrete image) are either hyperbolic or parabolic (the latter corresponding to loops are around the punctures; they are conjugate in $E(G)$ to one of the elements $E(C_1),\ldots,E(C_n)$).