Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds

Abstract

We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Z Γ , Σ o ( λ ) associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η ( D ) associated to the Dirac operator D over a convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 and that exp ( π i η ( D ) ) = Z Γ , Σ o ( 0 ) , thus extending Millson's formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil–Petersson metric on Schottky space.