According to **Uniformization theorem** every compact Riemann surface $\Sigma$ of genus $g\ge2$ is isomorphic to a space that can be obtained by the action of a Fuchsian group on upper half plane $\mathbb{H}$

$$\Sigma\simeq\frac{\mathbb{H}}{\Gamma}$$ 

On the Teichmüller or moduli space of such Riemann surfaces, one can consider **Fenchel-Nielsen coordinates** $\{\ell_a,\tau_a\}_{a=1}^{3g-3}$.

On the other hand, Selberg zeta function for a compact Riemann surface can be written as:

$$Z(s)\equiv\prod_{\{\gamma_p\}}\prod_{n}\left(1-e^{-(n+s)\ell_{\gamma_p}}\right)$$

In which $\{\gamma_p\}$ is the set of primitive elements of Fuchsian group $\Gamma$ and $\ell_{\gamma_p}$ is the length of the corresponding *simple closed geodesic* with respect to hyperbolic metric on $\Sigma$ induced from Poincare metric on $\mathbb{H}$. There are several questions:

 - What is the number of primitive elements of $\Gamma$? 
(There are supposedly infinitely many simple closed geodesics on $\Sigma$ so it should be infinite.)
 - What is the number of generators of $\Gamma$?
 - Is there any relation between number of generators of $\Gamma$ and genus of the surface? 
 - There are some quantities on the Riemann surface that can be expressed in terms of Selberg zeta function. For example **Determinant of Laplacian acting on various tensor fields on the Riemman surface** can be written in terms of Selberg zeta function. These quantities are thus dependent to the Teichmüller/moduli parameters. What is the relation between lengths $\{\ell_{a}\}$, the Fenchel-Nielsen coordinates and $\{\ell_{\gamma_p}\}$ in Selberg zeta function. 

Regarding the last question, it seems that we can consider $3g-3$ primitive elements $\{\hat{\gamma}_{p_i}\}_{i=1}^{3g-3}$ of $\Gamma$ and identify their lengths with Fenchel-Nielsen length coordinates $\{\ell_a\}_{a=1}^{3g-3}$:

$$\ell_{\hat{\gamma}_{p_i}}\equiv\ell_i \qquad i=1,\cdots,3g-3$$

If that is the case:
 
- What is the role of other lengths? Are they related to the set $\{\hat{\gamma}_p\}$ by the action of Mapping class group?