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For a finite-index subgroup of the modular group, the quotient of the complex upper half-plane becomes a compact Riemann surface after adjoining its cusp classes. At an elliptic fixed point use the quotient coordinate for the finite stabilizer; at a cusp use its exponential local parameter. Weight-zero modular forms descend to holomorphic functions on this compact surface, and hence are constant by the maximum modulus principle.

The full spectrum of an element is its spectrum together with every bounded component of its complement. Its complement is the unbounded resolvent component, and the maximum modulus principle controls a holomorphic function on each filled hole by its values on the spectral boundary.

If a function is continuous on the closure of a bounded plane domain and holomorphic inside, the compact closure supplies a point of maximum modulus. Unless the function is constant, the maximum modulus principle places every maximum on the boundary.