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

TL;DR

This paper extends the analysis of Selberg zeta functions and eta invariants to odd-dimensional convex co-compact hyperbolic manifolds, linking spectral theory, scattering theory, and geometric invariants.

Contribution

It introduces a meromorphic extension of the Selberg zeta function of odd type and relates the eta invariant of the Dirac operator to this zeta function, extending Millson's formula.

Findings

Proves the meromorphic extension of the Selberg zeta function of odd type.

Establishes the relation between eta invariant and the zeta function at zero.

Analyzes spectral and scattering theory of the Dirac operator on hyperbolic manifolds.

Abstract

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type $Z_{\Gamma,\Sigma}^{\rm o}(\lambda)$ associated to the spinor bundle $\Sigma$ on odd dimensional convex co-compact hyperbolic manifolds $X:=\Gamma\backslash\hh^{2n+1}$. We define a natural eta invariant $\eta(D)$ associated to the Dirac operator $D$ on $X$ and prove that $\eta(D)=\frac{1}{\pi i}\log Z_{\Gamma,\Sigma}^{\rm o}(0)$, thus extending Millson's formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant for the odd signature operator and, under some conditions, we describe it on the Schottky space of 3-dimensional Schottky hyperbolic manifolds and relate it to Zograf factorization formula.