Suppose $\Lambda$ is a finite (more than $2$ points) or countable discrete set in the unit disk $\mathbb{D}\subset\mathbb{C}$, how does one generate the Fuchsian group $\Gamma$ for which $\mathbb{D}/\Gamma\cong \mathbb{D}\backslash\Lambda$.
For finite sets, for example, is it simply a free product of cyclic groups generated by parabolic elements?