Noncongruence subgroups and Maass waveforms

The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke–Selberg and Phillips–Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of “new” discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) certify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting