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The study of spectral theory of finite-area hyperbolic surfaces is intimately related to number theory, in particular by the importance of Maass cusp forms. The counting of resonances is of importance, e.g., for quantum chaos.

What are motivations to study spectral theory of geometrically finite infinite-area hyperbolic surfaces? Are there applications in number theory as well?

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  • $\begingroup$ Obviously - thin subgroups comes to mind... See all the affine sieve works for example (Bourgain-Gamburd-Sarnak,Meiri-Sarnak,Meiri-Lubotzky,Lubotzky-Rosenzweig and many others). $\endgroup$
    – Asaf
    Oct 21, 2013 at 20:49
  • $\begingroup$ P.S. maybe the most important reference in the field - Lax-Phillips - "The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces", grew out from a point-counting problem, as the name of the article suggests. $\endgroup$
    – Asaf
    Oct 21, 2013 at 21:19

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