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.


2 Answers 2


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.


Yes, it is the standard material. The group is a discrete group of fractional-linear transformations of the upper half-plane, freely generated by $n-1$ parabolic transformations. There are many books discussing this, for example, L. R. Ford, Automorphic functions. Finding the group generators from the given points is a difficult problem: it is essentially an existence theorem, and the dependence of the generators on the points is complicated and highly transcendental. There are many detailed studies of small cases, for example the first non-trivial case $n=4$ and all points real. This is called the "problem of accessory parameters" for the Heun equation. You may look at the work of Zograf and Takhtajan on this.

EDIT. After having finished my answer I noticed the answer by Francesco Polizzi, and I have to apologize for implying that the case $n=3$ is "trivial". It depends on the point of view. The problem of accessory parameters is trivial (there are none). So we have a single group and a single function in this case. Still this important group and this function are not trivial and they are subject of research.


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