Abstract: Motivated by the question of understanding topological invariants of hyperbolic three-manifolds arising from Floer theory, I will discuss the problem of determining the spectra of Dirac operators on hyperbolic three-manifolds associated with flat spin^c connections. In particular, I will focus on the case of first Betti number one: in joint work with M. Lipnowski we introduce a method (taking as input natural quantities arising in hyperbolic geometry) which determines in concrete examples the locus corresponding to Dirac operators with kernel.