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On the functional equation of twisted Ruelle zeta function and Fried's conjecture

Jay Jorgenson, Min Lee, Lejla Smajlovic

Abstract

Let $M$ be a finite volume hyperbolic Riemann surface with arbitrary signature, and let $χ$ be an arbitrary $m$-dimensional multiplier system of weight $k$. Let $R(s,χ)$ be the associated Ruelle zeta function, and $\varphi(s,χ)$ the determinant of the scattering matrix. We prove the functional equation that $R(s,χ)\varphi(s,χ) = R(-s,χ)\varphi(s,χ)H(s,χ)$ where $H(s,χ)$ is a meromorphic function of order one explicitly determined using the topological data of $M$ and of $χ$, and the trigonometric function $\sin(s)$. From this, we determine the order of the divisor of $R(s,χ)$ at $s=0$ and compute the lead coefficient in its Laurent expansion at $s=0$. When combined with results by Kitano and by Yamaguchi, we prove further instances of the Fried conjecture, which states that the R-torsion of the above data is simply expressed in terms of $R(0,χ)$.

On the functional equation of twisted Ruelle zeta function and Fried's conjecture

Abstract

Let be a finite volume hyperbolic Riemann surface with arbitrary signature, and let be an arbitrary -dimensional multiplier system of weight . Let be the associated Ruelle zeta function, and the determinant of the scattering matrix. We prove the functional equation that where is a meromorphic function of order one explicitly determined using the topological data of and of , and the trigonometric function . From this, we determine the order of the divisor of at and compute the lead coefficient in its Laurent expansion at . When combined with results by Kitano and by Yamaguchi, we prove further instances of the Fried conjecture, which states that the R-torsion of the above data is simply expressed in terms of .
Paper Structure (33 sections, 15 theorems, 208 equations)

This paper contains 33 sections, 15 theorems, 208 equations.

Table of Contents

  1. Introduction
  2. R-torsion and analytic torsion
  3. Ruelle zeta function and the Fried conjecture
  4. The purpose of this article
  5. Comparison to some recent results
  6. Outline of the article
  7. Preliminaries
  8. Basic notation
  9. Unitary multiplier systems
  10. Action on parabolic representatives
  11. Action on primitive elliptic elements
  12. The Gamma and related functions
  13. Scattering determinant
  14. The Selberg zeta function and Ruelle zeta function
  15. Zeta regularized determinants
  16. ...and 18 more sections

Key Result

Lemma 2.1

Let $\Gamma\subset \mathrm{SL}(2,\mathbb{R})$ be a Fuchsian group of the first kind of genus $g\geq 1$ which does not contain parabolic elements and such that $-I_2\in\Gamma$. Assume that $\Gamma$ contains $\rho\geq 1$ classes of elliptic elements, with representatives $R_1,\ldots,R_\rho$ of orders

Theorems & Definitions (36)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • Remark 3.2
  • proof
  • Proposition 3.3
  • proof
  • Corollary 3.4
  • ...and 26 more