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

Author: 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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Keywords: Maass cusp form, embedded eigenvalues, Hecke triangle group, hyperbolic surfaces
Article copyright: © Copyright 1995 American Mathematical Society