[Paper Review] Fried conjecture in small dimensions

Nguyen Viet Dang, Colin Guillarmou+2 more|HAL (Le Centre pour la Communication Scientifique Directe)|Jul 3, 2018

Advanced Topology and Set Theory4 citations

TL;DR

This paper proves the Fried conjecture relating the analytic torsion and the value at zero of the twisted Ruelle zeta function for smooth Anosov vector fields on flat vector bundles in dimension 3. It establishes local constancy of the zeta function at zero under a spectral condition in higher dimensions and confirms the conjecture for Anosov flows near the geodesic flow on hyperbolic 3-manifolds, providing the first non-analytic examples where the conjecture holds in variable negative curvature.

ABSTRACT

We study the twisted Ruelle zeta function $ζ_X(s)$ for smooth Anosov vector fields $X$ acting on flat vector bundles over smooth compact manifolds. In dimension $3$, we prove Fried conjecture, relating Reidemeister torsion and $ζ_X(0)$. In higher dimensions, we show more generally that $ζ_X(0)$ is locally constant with respect to the vector field $X$ under a spectral condition. As a consequence, we also show Fried conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic $3$-manifolds. This gives the first examples of non-analytic Anosov flows and geodesic flows in variable negative curvature where Fried conjecture holds true.

Motivation & Objective

• To prove the Fried conjecture relating analytic torsion and the value at zero of the twisted Ruelle zeta function for Anosov vector fields on flat vector bundles.
• To extend the validity of the Fried conjecture to non-analytic Anosov flows and geodesic flows in variable negative curvature.
• To establish local constancy of the twisted Ruelle zeta function at zero under a spectral condition in higher dimensions.
• To confirm the conjecture for Anosov flows near the geodesic flow on the unit tangent bundle of hyperbolic 3-manifolds.
• To provide the first examples of non-analytic Anosov flows where the Fried conjecture holds true in the context of variable negative curvature.

Proposed method

• Analyzes the twisted Ruelle zeta function ζ_X,ρ(λ) defined via primitive closed orbits of a smooth Anosov vector field X on a compact manifold M with flat Hermitian bundle E.
• Applies meromorphic continuation techniques for the zeta function, building on prior work by Rugh, Fried, and Giulietti-Liverani-Pollicott for C^∞ Anosov flows.
• Uses the spectral condition that the twisted Laplacian has no zero modes in the unstable and stable directions to establish local constancy of ζ_X(0) with respect to X.
• Relies on the Ray-Singer analytic torsion τ_ρ(M) and its equivalence to Reidemeister torsion via Cheeger-Müller theorem for acyclic unitary representations.
• Applies Selberg trace formula techniques and representation theory of SO(n) to analyze the zeta function on trace-free symmetric tensors in hyperbolic manifolds.
• Establishes the order of zeros of the Selberg zeta function using the kernel dimensions of twisted Laplacians and divergence operators.

Experimental results

Research questions

• RQ1 Does the Fried conjecture hold for Anosov flows in dimension 3 with flat vector bundles?
• RQ2 Can the twisted Ruelle zeta function be shown to be locally constant at s=0 under a spectral condition in higher dimensions?
• RQ3 Does the Fried conjecture extend to geodesic flows on hyperbolic 3-manifolds with variable negative curvature?
• RQ4 Are there non-analytic Anosov flows for which the Fried conjecture holds true?
• RQ5 What is the precise relation between the order of zeros of the Selberg zeta function and the kernel of twisted Laplacians on trace-free symmetric tensors?

Key findings

• The Fried conjecture is proven in dimension 3 for smooth Anosov vector fields on flat vector bundles, showing |ζ_X,ρ(0)|^{(-1)^{n_0}} = τ_ρ(M).
• In higher dimensions, ζ_X(0) is shown to be locally constant with respect to the vector field X under a spectral condition on the twisted Laplacian.
• The conjecture is verified for Anosov flows near the geodesic flow on the unit tangent bundle of a hyperbolic 3-manifold, even in variable negative curvature.
• This provides the first known examples of non-analytic Anosov flows where the Fried conjecture holds true.
• The order of zeros of the Selberg zeta function Z_{S,σ_m}(s) is computed via kernel dimensions of twisted Laplacians and divergence operators.
• For s₀ ≠ n/2, the order of zero is dim(ker(∇*∇ - n²/4 - m + (s₀ - n/2)²) ∩ ker D*), and twice that for s₀ = n/2.