Fundamental domain (reviewed)

If $G\subseteq\Gamma$ is a subgroup of the modular group, then a closed set $F\in\mathcal{H}\cup\mathbb{Q}\cup\{\infty\}$ is said to be a fundamental domain for $G$ if:

1. For any point $z\in\mathcal{H}$ there is a $g\in G$ such that $gz\in F$.
2. If $z\not=z'\in F$ are equivalent with respect to the action of $G$, that is, if $z'=gz$ for some $g\in G$, then $z$ and $z'$ belong to $\partial F$, the boundary of F.