On a special value of the Ruelle L-function

TL;DR

This paper establishes a precise relationship between the value of the Ruelle L-function at zero and the Franz-Reidemeister torsion for certain hyperbolic threefolds with a unitary local system, linking spectral and topological invariants.

Contribution

It proves that for hyperbolic threefolds with a specific local system, the Ruelle L-function's value at zero equals the square of the Franz-Reidemeister torsion, under the condition of vanishing first cohomology.

Findings

Ruelle L-function at s=0 equals the square of Franz-Reidemeister torsion

Valid for hyperbolic threefolds with one cusp and rank-one local systems

First cohomology vanishing is a key condition

Abstract

For a unitary local system of rank one on a complete hyperbolic threefold of finite volume which has only one cusp, the Ruelle L-function is defined. We will show that if the first cohomology group of the local system vanishes its value at s=0 is equal to the square of the Franz-Reidemeister torsion.