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 and their relations with Selberg zeta functions. Using the Selberg trace formula and a detailed analysis of the unipotent orbital integral, we show that the eta and zeta functions defined by the relative traces are regular at the origin so that we can define the eta invariant and the regularized determinant. We also show that the Selberg zeta function of odd type has a meromorphic extension over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], prove a relation of the eta invariant and a certain value of the Selberg zeta function of odd type, and derive a corresponding functional equation. These results generalize the earlier work of John Millson to hyperbolic manifolds with cusps. We also prove that the Selberg zeta function of even type has a meromorphic extension over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /], relate it to the regularized determinant, and obtain a corresponding functional equation.