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

Hurwitz spaces are spaces of pairs S, f where S is a Riemann surface and f : S → ̂ C a meromorphic function. In this work, we study 1-dimensional Hurwitz spaces Hp of meromorphic p-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of p − 1 /2 transpositions and the monodromy group is the dihedral group Dp. We prove that the completion Hp of the Hurwitz space Hp is uniformized by a non-nomal index p 1 subgroup of a triangular group with signature 0; p, p, p . We also establish the relation of the meromorphic covers with elliptic functions and show that Hp is a quotient of the upper half plane by the modular group Γ 2 ∩ Γ0 p . Finally, we study the real forms of the Belyi projection Hp → ̂ C and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated.