Billiard flows, Laplace eigenfunctions, and Selberg zeta functions

Eigenfunctions of Laplace operators on Riemannian manifolds and orbifolds are objects of common interest in various fields, e.g. number theory, spectral theory, harmonic analysis, and mathematical physics. In particular by results from the latter field it is long known that these eigenfunctions are intimately related to geometric properties of the orbifold. However, the full extent of this relation and its consequences is still an active field of research.

In this talk we will consider the Hecke triangle surfaces L\H, where H is the hyperbolic plane and L a cofinite Hecke triangle group, and will concentrate on Maass cusp forms for L. By Selberg theory, the spectral parameters of these eigenfunctions are contained in the zeros of the Selberg zeta function for L\H. We will show how transfer operator approaches allow to lift this relation from the spectral level to one between the actual eigenfunctions and periodic geodesics. Moreover we will discuss the current status of reproving the connection between Maass cusp forms and Selberg zeta functions within this framework.

This results in a characterization of Maass cusp forms as solutions of finite-term functional equations and, simultaneously, a geometrically motivated notion of period functions. Further, it provides us with two new formulations of the Phillips-Sarnak conjecture on nonexistence of even Maass cusp forms.

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