27 september, Martin Olbrich (Technische Universität Clausthal and Georg-August-Universität Göttingen): Relations between length spectrum and topology of certain infinite volume hyperbolic manifolds

Abstract. It is known for a long time that the length spectrum of a finite volume hyperbolic surface determines its topology completely. This can be established as follows: the length spectrum defines a certain holomorphic function on a right half-plane, the Selberg zeta function. Using Selberg's trace formula one can meromorphically continue this zeta function to the whole complex plane. Moreover, one can analyze the order of its singularities. In particular, the Euler characteristic and the number of cusps of the surface can be recovered by these orders. This analysis becomes much more involved for infinite volume surfaces or even higher dimensional manifolds of infinite volume. The problem is to separate topological from spectral contributions to the singularities. We discuss how this problem is related to conjectures of Patterson. Eventually, we show that the Euler characteristic of a geometrically finite surface without cusps is determined by its length spectrum.