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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 $\mathcal{M}_k$.

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If you want to see anything more specific than Lee's answer, you learn that this is the infamous "accessory parameters problem". –  Misha May 22 '13 at 23:45
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1 Answer 1

One can always find a fundamental domain for $G$ which is an ideal polygon $P \subset \mathbb{H}$ having $2k-2$ vertices at infinity and $2k-2$ sides, so that the sides are written in cyclic order around $P$ as $a_1 a_2 \ldots a_{k-1} \bar a_{k-1} \ldots \bar a_2 \bar a_1$, and so that the group $G$ is generated by $k-1$ side pairings which are isometries identifying $a_i$ to $\bar a_i$ for $i=1,\ldots,k-1$. This gives a fairly explicit description of $G$, expressed in the language of the Poincare polygon theorem.

One can indeed use this idea, or closely related ideas, to parameterize $\cal M_k$: see the paper of Epstein and Bowditch, "Natural triangulations associated to a surface" Topology 27 (1988); and the paper of Penner, "The decorated Teichmüller space of punctured surfaces" Comm. Math. Phys. 113 (1987).

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Sure, but how do the matrices corresponding to these isometries look like? Is it discussed in those references? –  Mohammad F. Tehrani May 23 '13 at 15:08
    
The derivation of the matrices should be covered in any textbook on hyperbolic geometry. In outline, if you fix the polygon $P$ in the upper half plane model, each side pairing $a_i \mapsto \bar a_i$ takes the endpoints of $a_i$ to the endpoints of $\bar a_i$. Once the image of a third point at infinity is determined, the matrix is determined. There is also a completeness condition for each cusp: the monodromy around that cusp must be parabolic. With that, you get a very explicit set of formulas parameterizing the matrices. –  Lee Mosher May 23 '13 at 16:45
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Lee: What Mohammad Tehrani wants is an "explicit" map from the tuple $(z_1,...,z_n)$ to the matrices generating the uniformizing Fuchsian group. This is essentially the "accessory parameters" problem which perplexed Klein and Poincare, and to which no satisfactory solution is known to this day, even if one allows infinite series as "explicit" solutions. The best results are due to A.Venkov (real case, he even wrote an explicit infinite series!), Zograf and Takhtajian. Needless to say, none of this is needed for constructing coordinates on the moduli space. –  Misha May 25 '13 at 12:57
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