Fundamental region of a modular subgroup (${\bigcup }_{\alpha \in \Gamma \backslash \Gamma (1)}\alpha \mathcal{F}$)

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.