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

$\displaystyle (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:

$\displaystyle (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)$).

Pablo Ares Gastesi 2005-08-31