Explicit Constructions of Finite Groups as Monodromy Groups

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 $G \simeq \operatorname{Mon}(\pi)$.