One-Dimensional Hurwitz Spaces, Modular Curves, and Real Forms of Belyi Meromorphic Functions

Hurwitz spaces are spaces of pairs where is a Riemann surface and a meromorphic function. In this work, we study -dimensional Hurwitz spaces of meromorphic -fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of transpositions and the monodromy group is the dihedral group . We prove that the completion of the Hurwitz space is uniformized by a non-nomal index subgroup of a triangular group with signature . We also establish the relation of the meromorphic covers with elliptic functions and show that is a quotient of the upper half plane by the modular group . Finally, we study the real forms of the Belyi projection and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.