### 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.