Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups

TL;DR

This paper establishes a connection between the absolute value at zero of the Ruelle zeta function for geodesic flows and the higher-dimensional Reidemeister torsion for hyperbolic orbifolds, using integral expressions derived from Selberg zeta functions.

Contribution

It provides a new integral expression for the Ruelle zeta function and links it to Reidemeister torsion, revealing asymptotic behavior influenced by the identity element.

Findings

Absolute value at zero of Ruelle zeta equals higher-dimensional Reidemeister torsion

Integral expression derived from Selberg zeta functional equation

Asymptotic behavior determined by identity element contribution

Abstract

We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in PSL(2;R). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.