Let $X$ be the $\mathbb{C}\mathbb{P}^1$ with $n$ points deleted. Let $n\geq 3$. If I understand correctly, the universal covering of $X$ is isomorphic to the upper half plane as a complex analytic space. 



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> **Q.** How one can describe the group of deck transformations of the universal covering as a subgroup of $\mathrm{PGL}(2,\mathbb{R})$?
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I expect this should be a standard material. A reference would be helpful.



Universal covering of a 2-sphere without $n$ points
I think this is part of the classical theory of Fuchsian groups.


For instance, in the case $n=3$ the fundamental group of the thrice punctured sphere can be explicitly identified with the congruence subgroup of level two $\Gamma(2) \subset \mathrm{PSL}(2, \, \mathbb{Z})$, see Theorem 2.34 in the book


 E. Girondo, G. González-Diez: *Introduction to compact Riemann surfaces and dessins d’enfants*, London Mathematical Society Student Texts **79** (2012). [ZBL1253.30001](https://zbmath.org/?q=an:1253.30001).


