Some Applications of Number Theory to the Theory of 3-Manifolds

In this talk, I will focus on hyperbolic 3-manifolds which are "arithmetic". These manifolds
provide a rich source of (counter-)examples for many topological phenomena. Moreover,
their arithmetic nature allows one to use results from number theory and the theory of
automorphic representations to deduce interesting results of topological nature, for
example, related to the Virtual Betti Number conjecture and the Virtual Fibration conjecture.

In the first half of the talk, I will present the basic notions. As the talk is intended for an
audience of topologists, only a minimal background in the number theory will be assumed.
The second part will be devoted to exploiting the arithmetic nature of these manifolds to
deduce results of interest for 3-manifold theory. The overall goal of this talk is to raise
interest and stimulate interaction between the number theorists and the topologists of
the MPI.