Eta Invariants and Regularized Determinants for Odd Dimensional Hyperbolic Manifolds with Cusps

We study eta invariants of Dirac operators and regularized determinants of Dirac Laplacians over hyperbolic manifolds with cusps. We follow Werner Müller (see [18], [19]) and use relative traces to define the eta function and the zeta function. We show regularities of eta and zeta functions at the origin so that we can define the eta invariant and the regularized determinant. By the Selberg trace formula, we show an equality between the eta invariant and the Selberg zeta function of odd type. The complete analysis of the unipotent orbital integral allows us to show the vanishing of the unipotent contribution in this relation. We also show that the eta invariant and the Selberg zeta function of odd type satisfy certain functional equation. These results generalize the earlier work of John Millson (see [12]) to hyperbolic manifolds with cusps. We also get the corresponding relation and the functional equation for the regularized determinant and the Selberg zeta function of even type where the unipotent factor plays a non-trivial role.