For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write

$$ P^1 \setminus S \cong \mathbb{H}/G $$

where $\mathbb{H}$ is the upper-half plane and $G\subset PSL(2,\mathbb{R})$ is a representation of $\pi_1(\mathbb{P}^1\setminus S)$. G can be generated by $(k-1)$-elements.

Is there an explicit description of how does G look like? In that case, the paremeters describing such G can give coordinates on $\mathcal{M}_k$.

For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write

$$ P^1 \setminus S \cong \mathbb{H}/G $$

where $\mathbb{H}$ is the upper-half plane and $G\subset PSL(2,\mathbb{R})$ is a representation of $\pi_1(\mathbb{P}^1\setminus S)$. G can be generated by $(k-1)$-elements.

Is there an explicit description of how $G$ looks? In that case, the parameters describing such $G$ can give coordinates on $\mathcal{M}_k$.