The Taylor expansion of Ruelle L-function at the origin and the Borel regulator

TL;DR

This paper investigates the properties of the Ruelle L-function on hyperbolic manifolds, establishing a functional equation, an analog of the Riemann hypothesis, and connecting its Laurent expansion coefficients to the Borel regulator in algebraic K-theory.

Contribution

It proves the functional equation and Riemann hypothesis analog for the Ruelle L-function, computes its Laurent expansion at zero, and links coefficients to the Borel regulator.

Findings

Ruelle L-function satisfies a functional equation.

The Laurent expansion at zero has coefficients related to volume and Borel regulator.

In three dimensions, the leading coefficient is explicitly identified.

Abstract

We will prove that Ruelle L-function for a cuspidal local system on an odd dimensional hyperbolic manifold with finite volume satisfies a functional equation and an analog of the Riemann hypothesis. We will also compute its Laurent expansion at the origin and will prove that the second coefficient coincides with a rational multiple of the volume up to a certain contribution from cusps. Moreover if the dimension is three we will identify the leading coefficient. Both of them will be intepreted by the Borel regulator in algebraic K-theory. Also a relation with the L^2-torsion will be discussed.