Uniformizations of the bordered/punctured Riemann surfaces

Question

The uniformization theorems of Riemann surfaces state that any Riemann surface can be constructed by an action of some group on some space. It is quite hard to find materials relating different uniformizations in the math literature. I have several questions related to different uniformizations. I appreciate any comment or pointing to references.

For a closed Riemann surface $\Sigma_{g,n}$ of genus $g$ with $n$ number of punctures:

• Is there a Fuchsian uniformization in terms of a Fuchsian group $\Gamma$ acting on the upper half-plane ${\mathbb{H}}$?

  I suppose there is a unique uniformization up to the action of $\text{PSL}(2,\mathbb{R})$, the group of orientation-preserving automorphisms of $\mathbb{H}$.

• Is there a Schottky uniformization in terms of a Schottky group $S$?

  I suppose there is but I am not sure about the uniqueness. Any comment would help.

• Is there a way to get Schottky uniformization from a Fuchsian one or vice versa?

• Is there a way to get objects like period matrix, prime form, abelian differential, and ... in terms of Fenchel-Nielsen coordinates?

• Is there a way to get complex analytic structure of Teichmuller in terms of Fenchel-Nielsen coordinates?

And the same set of questions for a bordered surfaces,

For a bordered Riemann surface $\Sigma_{g,n}^{(L_1,\cdots,L_n)}$ of genus $g$ with $n$ number of boundaries of lenghts $\{L_1,\cdots,L_n\}$:

• Is there a Fuchsian uniformization in terms of a Fuchsian group $\Gamma$ acting on the upper half-plane ${\mathbb{H}}$?

  I suppose there is a unique uniformization up to the action of $\text{PSL}(2,\mathbb{R})$, the group of orientation-preserving automorphisms of $\mathbb{H}$.

• Is there a Schottky uniformization in terms of a Schottky group $S$?

• Is there a way to get Schottky uniformization from a Fuchsian one or vice versa?

• Is there a way to get objects like period matrix, prime form, abelian differential, and ... in terms of Fenchel-Nielsen coordinates?

• Is there a way to get complex analytic structure of Teichmuller in terms of Fenchel-Nielsen coordinates?

  I am not even sure that in this case, the Teichmuller space has a complex analytic structure.

Answer 1

In the closed or bounded case, there is a uniformization of a Riemann surface by a fuchsian group. In the closed case this is the content of the uniformization theorem. The bounded case follows from a standard doubling trick.

Classical Schottky groups are always closed, so they cannot uniformize Riemann surfaces with boundary. All closed Riemann surfaces are uniformized by Schottky groups - I believe that the correct reference is

Maskit, A characterisation of Schottky groups, J. Analyse Math. 19 (1967) 227–230

It is "straight-forward" to convert a Schottky or fuchsian uniformization into a Riemann surface. So, if you have a black box that can solve the fuchsian or Schottky uniformization problem, then you can convert in one direction or the other.

Similarly, the FN coordinates specify a Riemann surface, so we can in principle deduce a period matrix. I don't know what the period matrix is for a bounded surface. I don't know what the prime form is.

A Riemann surface (closed or bounded) does not have a "canonical" abelian differential. It also does not have a "canonical" quadratic differential (which is how I interpret your question about "Teichmüller structure").

Answer 2

In the case of compact Riemann surface of genus $g$ with $n$ marked points, I can give an answer to two of the questions I asked:

Is there a Fuchsian uniformization in terms of a Fuchsian group $\Gamma$ acting on the upper half-plane $\mathbb{H}$?

Answer :

There is a unique uniformization up to the action of $\text{PSL}(2,\mathbb{R})$, the group of orientation-preserving automorphisms of $\mathbb{H}$.

Is there a way to get Schottky uniformization from a Fuchsian one or vice versa?

Answer :

Let $\Gamma_F$ denoted the uniformizing group of Fuchsian uniformization and $\Gamma_S$ denoted the uniformizing group of Schottky uniformization. Then there are the following facts:

• $\Gamma_F$ acts on a fundamental polygon in upper half-plane $\mathbb{H}$. Compact Riemann surface $\Sigma_{g,n}$ can be viewed as

$$\Sigma_{g,n}\cong \frac{\mathbb{H}}{\Gamma_F}$$

• $\Gamma_S$ acts on the domain of discontinuity $\Omega_{\Gamma_S}$ of $\Gamma_S$. Compact Riemann surface $\Sigma_{g,n}$ can be viewed as

$$\Sigma_{g,n}\cong \frac{\Omega_{\Gamma_S}}{\Gamma_S}$$

Definition I : The limit set of the action of $\Gamma_S$ is the closure of the set of the attractive and repelling fixed points of the loxodromic elements $g\in\Gamma_s$.

Definition II : The domain of discontinuity $\Omega_{\Gamma_S}$ of the group $\Gamma_S$ is the complement of the limit set of the action of $\Gamma_S$ in Riemann sphere $\mathbb{C}^*$ ($\Omega_{\Gamma_S}\subset \mathbb{C}^*$).

Let $\{a_i\}_{i=1}^{n}$ denoted the set of generators of the fundamental group $\pi_1(\Sigma_{g,n})$ of the compact surface $\Sigma_{g,n}$, let $\mathcal{N}\equiv\langle a_1,\cdots,a_m\rangle$ be the normal subgroup of the $\pi_1(\Sigma_{g,n})$ which is generated by half of the genrators. Using the fact that $\Gamma_F\cong\pi_1(\Sigma_{g,n})$, the relation between Shottky group $\Gamma_S$ and Fuchsian group $\Gamma_F$ can be written as follows:

$$\Gamma_S\cong\frac{\Gamma_F}{\mathcal{N}}$$