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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. So the natural question is that 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?

  • Following Quillen, The determinant of Dirac operator is defined through the corresponding Laplacian which in turn can be expressed in terms of Selberg zeta function. Regarding the fact that FN coordinates can not give a complex structure to Teichmuller space, is there any notion of "holomorphic factorization" which makes it possible to define the determinant of Dirac operator directly in terms of Selberg zeta function?

If that is not the case,

  • How is it possible to obtain dependence of determinant of the Dirac operator on Fenchel-Nielsen coordinates?
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I can answer some of the questions: there are infinitely many primitive elements, the so called Prime Geodesic Theorem tells you how the number of primes with bounded length growth as the bound tends to infinity.

The number of generators is $2g$. This is elementary surface theory. There are generators $\gamma_1,\dots,\gamma_{2g}$ with the generating relation: $$ [\gamma_1,\gamma_2]\cdots[\gamma_{2g-1},\gamma_{2g}]=1, $$ where $[a,b]=aba^{-1}b^{-1}$.

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  • $\begingroup$ Thank you for your answer, so there are $2g-2$ independent generators. $\endgroup$ – QGravity Sep 11 '16 at 16:44
  • $\begingroup$ Well, no, it is not a free group. The first $2g-1$ generators generate a free subgroup, though. $\endgroup$ – user1688 Sep 11 '16 at 17:47
  • $\begingroup$ At least in this case, $\Gamma\subset{\bf {PSL}}(2,\mathbb{R})$, (${\bf (PSL)}(2,\mathbb{R}$) is the group of orientation preserving authomorphisms of upper half plane $\mathbb{H}$) and acts freely on ${\mathbb{H}}$. $\endgroup$ – QGravity Sep 11 '16 at 18:54

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