 Finiteness of isospectral hyperbolic surfaces

For a fixed closed hyperbolic surface, there are only finitely many isometry classes with its Laplacian spectrum. The Selberg trace formula gives a common discrete length spectrum and common positive hyperbolic systole. Mumford's compactness theorem confines this family to a compact subset of moduli space. If distinct members accumulated, choose nearby markings at an accumulation point and use finite length coordinates for hyperbolic surfaces. Each of these finitely many lengths converges while remaining in one discrete spectrum, so it is eventually constant. The length coordinates then force eventual equality, a contradiction. This finiteness theorem is weaker than the Wolpert generic spectral rigidity theorem.