Explicit Constructions of Finite Groups as Monodromy Groups

TL;DR

This paper provides two explicit constructive methods to realize any finite group as a monodromy group of a morphism of Riemann surfaces, expanding the understanding of the connection between finite groups and algebraic geometry.

Contribution

It offers two new constructive proofs of Greenberg's theorem, one using free groups and the other employing triangle groups, to explicitly realize finite groups as monodromy groups.

Findings

Constructive proof using free groups and their properties.

Explicit realization of finite groups via subgroups of triangle groups.

Provides concrete morphisms with specified monodromy groups.

Abstract

In 1963, Greenberg proved that every finite group appears as the monodromy group of some morphism of Riemann surfaces. In this paper, we give two constructive proofs of Greenberg's result. First, we utilize free groups, which given with the universal property and their construction as discrete subgroups of $\operatorname{PSL}_2(\mathbb{R})$, yield a very natural realization of finite groups as monodromy groups. We also give a proof of Greenberg's result based on triangle groups $\Delta(m, n, k)$. Given any finite group $G$, we make use of subgroups of $\Delta(m, n, k)$ in order to explicitly find a morphism $\pi$ such that $G \simeq \operatorname{Mon}(\pi)$.