= 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>.
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