[Paper Review] A special value of Ruelle L-function and the theorem of Cheeger and Muller

Ken‐ichi Sugiyama|ArXiv.org|Mar 14, 2008

Geometric and Algebraic Topology12 references3 citations

TL;DR

This paper establishes a Cheeger-Müller-type theorem for noncompact hyperbolic threefolds of finite volume, proving that the special value of the Ruelle L-function at zero equals the square of the product of modified Ray-Singer torsion and a geometric period. It explicitly computes this value for unitary local systems on knot complements using twisted Alexander polynomials, generalizing analytic torsion identities to noncompact settings.

ABSTRACT

We will show a theorem of a type of Cheeger and Muller for a noncompact complete hyperbolic threefold of finite vulume. As an application we will compute a special value of Ruelle L-function at the origin for a unitary local system which is cuspidal.

Motivation & Objective

• To extend the Cheeger-Müller theorem to noncompact complete hyperbolic threefolds of finite volume.
• To establish a formula relating the special value of the Ruelle L-function at zero to analytic and combinatorial invariants.
• To compute the special value of the Ruelle L-function for cuspidal unitary local systems on knot complements.
• To generalize analytic torsion identities to noncompact manifolds with cusps using spectral and Reidemeister torsion techniques.

Proposed method

• Uses J. Park's result that the Ruelle L-function has a zero of order $2h^1(X,\rho)$ at $z=0$, reducing the problem to analyzing the leading coefficient.
• Applies the spectral zeta function $\zeta_X(z,\rho)$ and relates its derivative at zero to the special value of the Ruelle L-function.
• Relies on the equivalence of Ray-Singer and Franz-Reidemeister torsions in the noncompact case, proven via limit arguments on compactifications with torus boundaries.
• Uses Fox calculus to compute twisted Alexander polynomials $\Delta_{K,\rho}(t)$ from the presentation of the knot group.
• Applies the long exact sequence in cohomology to relate the vanishing of $h^1(X_K,\rho)$ to non-vanishing of $\Delta_1(1)$ and $A_K(\xi)\neq 0$
• Computes the special value $R_{X_K}(0,\rho) = \left| \frac{A_K(\xi)}{1-\xi} \right|^2$ using the relation $\tau(X_K,\rho) = |\Delta_{K,\rho}(1)|$ when $h^1=0$.

Experimental results

Research questions

• RQ1 Does the Cheeger-Müller theorem hold for noncompact hyperbolic threefolds with cusps?
• RQ2 Can the special value of the Ruelle L-function at zero be expressed in terms of analytic and combinatorial torsion invariants?
• RQ3 What is the explicit value of the Ruelle L-function at zero for a unitary local system on a knot complement?
• RQ4 How does the twisted Alexander polynomial relate to the special value of the Ruelle L-function?
• RQ5 What conditions ensure the vanishing of $h^1(X_K,\rho)$, and how does this affect the L-function value?

Key findings

• The special value of the Ruelle L-function at zero is given by $\lim_{z\to 0} z^{-2h^1(X,\rho)} R_X(z,\rho) = (\tau^*(X,\rho) \cdot \mathrm{Per}(X))^2$, where $\tau^*$ is modified Ray-Singer torsion and $\mathrm{Per}(X)$ is a geometric period.
• When $h^1(X,\rho) = 0$, the special value simplifies to $R_X(0,\rho) = \tau(X,\rho)^2$, with $\tau$ being the standard Franz-Reidemeister torsion.
• For a hyperbolic knot $K$ and a unitary character $\rho_\xi$ with $\xi \neq 1$, the special value is $R_{X_K}(0,\rho) = \left| \frac{A_K(\xi)}{1-\xi} \right|^2$, where $A_K(t)$ is the Alexander polynomial.
• The vanishing of $h^1(X_K,\rho)$ is equivalent to $A_K(\xi) \neq 0$ and $\xi \neq 1$, ensuring the special value is computable.
• The proof relies on the limit of compactifications with non-product metrics, using Dai-Fang's formula to show the equality of Ray-Singer and Franz-Reidemeister torsions in the limit.
• The twisted Alexander polynomial $\Delta_{K,\rho}(t)$ satisfies $\Delta_{K,\rho}(1) = \frac{A_K(\xi)}{1-\xi}$, linking it directly to the special value.