Standard fundamental domain of the modular group ($\mathcal{F}$)

The standard fundamental domain isEvery orbit of the modular group meets $\mathcal{F}$.

For a finite-index subgroup of the modular group, take the union of translates of the standard fundamental domain of the modular group over left coset representatives. Boundary identifications are inherited from the subgroup action. The truncated region is compact, and its finitely many cusp ends have exponential parameters; this proves compactness of the compactified modular curve and boundedness of invariant cusp-form norms.

For the boundary convention $-1/2<\mathrm{Re}z\le 1/2$, retaining only the right half of the unit circle, the exceptional points are $i$ and ${\rho }_{+}=e^{i\pi /3}$. Their stabilizers in $S{L}_2(\mathbb{Z})$ are the cyclic groups generated by $\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$ and $\left(\begin{array}{cc}1& -1\\ 1& 0\end{array}\right)$, of orders four and six. Every other stabilizer is $\{I,-I\}$.

Given $\tau \in \mathfrak{h}$, choose a primitive pair $(c,d)$ minimizing $|c\tau +d|$ and apply a modular matrix with bottom row $(c,d)$. This maximizes the imaginary part within the orbit. Translation puts the real part in $[-1/2,1/2]$, and inversion then shows that the imaginary part is at least $\sqrt{3}/2$.