I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related:
On the one hand, there is a parametrization coming from hyperbolic geometry. We can think of the moduli space of a surface $S$ as the quotient of its Teichmuller space by the mapping class group: $\mathcal{M}(S) = \mathcal{T}(S)/\mathrm{MCG}(S)$. The Teichmuller space can be parametrized using e.g. Fenchel-Nielsen coordinates or Penner’s $\lambda$-lengths.
On the other hand, there are parametrizations coming from thinking of the surface as a complex manifold, for instance in the case of a punctured sphere we use the location of the punctures modulo bi-holomorphisms of the surface. On higher genus surfaces we consider holomorphic differentials on the surface and their periods.
But how can we relate the pictures?
Let me give a simple example where I understand things perfectly, and which I would like to understand if and how generalizes. Consider the case where $S$ is a disk with $n$ marked points on the boundary. Since the mapping class group is trivial, $\mathcal{M}(S)=\mathcal{T}(S)$. The moduli space can be thought of as $$\mathcal{M}(S) = \left\{ (\sigma_1, \dots, \sigma_n) \in \mathbb{R}^n | \sigma_i \neq \sigma_j \mathrm{\ if\ } i \neq j \right\}/\mathrm{SL}(2,\mathbb{R}),$$ While the Teichmuller space can be described using Penner’s $\lambda$-lengths as $$\mathcal{T}(S)= \{\lambda_{i,j} \in \mathbb{R}^+ | \lambda_{i,j} \lambda_{k,l} = \lambda_{i,l} \lambda_{k,j} + \lambda_{i,k} \lambda_{j,l}, \mathrm{\ if \ } e_{(i,j)} \cap e_{k,l} \neq \emptyset \}/(\lambda_{i,j} \sim t_i \lambda_{i,j}, \forall\ i,j),$$ Where $e_{i,j}$ denote an arc joining the marked points with labels $(i,j)$. The two descriptions can be shown to be equivalent via the identification $\lambda_{i,j} = \sigma_i - \sigma_j$.
Are there similar descriptions for other surfaces?
There are two cases I would like to understand.
When $S$ is a punctured sphere, I would naively expect something like $(\sigma_i - \sigma_j) \sim \ell_{i,j} \exp\left(2 \pi i \frac{\tau_{i,j}}{\ell_{i,j}}\right)$, where $(\ell_{i,j}, \tau_{i,j})$ are Fenchel-Nielsen coordinates associated to a curve separating punctures $i$ and $j$ from the other ones. I would expect so because the limit $\ell \to 0$ seems to coincide with the limit $\sigma_i \to \sigma_j$ and because as we move the twisting parameter $\tau_{i,j}$ the punctures are rotating around each other
When $S$ is a torus we have the famous description $\mathcal{M}(S) = \left\{ y \in \mathbb{C} |\ \mathrm{Im}(y) > 0 \right\}/\mathrm{SL}(2,\mathbb{Z})$, coming from taking ratios of the periods of the holomorphic differential on $S$. Since we have a single pair of Fenchel-Nielsen coordinates I would naively expect $y = \tau + i \ell/\tau$, but it seems to me that the mapping class group acts differently on $y$ and on $\tau + i \ell/\tau$
I would be very happy to have some references which explore at least one of the above two examples in the most explicit way possible.
