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