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ISSN 1088-6834(e) ISSN 0894-0347(p)

On the existence of Maass cusp forms on hyperbolic surfaces with cone points

Author(s): Christopher M. Judge
Journal: J. Amer. Math. Soc. 8 (1995), 715-759.
MSC: Primary 11F72; Secondary 58G25
MathSciNet review: 1273415
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Abstract: The perturbation theory of the Laplace spectrum of hyperbolic surfaces with conical singularities belonging to a fixed conformal class is developed. As an application, it is shown that the generic such surface with cusps has no Maass cusp forms ( ${L^2}$ eigenfunctions) under specific eigenvalue multiplicity assumptions. It is also shown that eigenvalues depend monotonically on the cone angles. From this, one obtains Neumann eigenvalue monotonicity for geodesic triangles in ${{\mathbf{H}}^2}$ and a lower bound of $\frac{1}{2}{\pi ^2}$ for the eigenvalues of `odd' Maass cusp forms associated to Hecke triangle groups.

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