Twisted Pollicott--Ruelle resonances and zeta function at zero on surfaces
Tristan Humbert, Zhongkai Tao
TL;DR
This work analyzes the zero of twisted Ruelle zeta functions for Anosov surface geodesic flows twisted by finite-dimensional representations. It introduces a Zariski-open set Ug of irreducible representations for which the order of vanishing at zero is fully controlled: m(g,ρ) = dim(ρ)(2G-2) if ρ factors through pi1(Σ), and m(g,ρ) = 0 otherwise; when ρ is acyclic and does not factor, zeta(g,ρ)(0) matches the Reidemeister torsion up to sign, extending Fried-type conjectures to generic non-unitary twists. The core method translates the problem into twisted Pollicott–Ruelle resonances, proving analytic dependence under perturbations and showing the generic absence of zero Jordan blocks for k=0,2, while allowing Jordan blocks in certain non-unitary cases. The paper also provides explicit computations in the Ad case via Selberg theory and constructs a non-factor irreducible tau with zero Jordan blocks tied to the Laplacian spectrum, highlighting subtle interactions between dynamics, topology, and spectral theory with potential extensions to broader flows and representations.
Abstract
For an orientable closed surface $(Σ,g)$ of genus $G$ with Anosov geodesic flow, we show the existence of an open subset $U_g$ of finite-dimensional irreducible representations of the fundamental group of its unit tangent bundle, whose complement has complex codimension at least one and such that for any $ρ\in U_g$, the twisted Ruelle zeta function $ζ_{g,ρ}(s)$ vanishes at $s=0$ to order ${\rm dim}(ρ)(2G-2)$ if $ρ$ factors through $π_1(Σ)$, and does not vanish otherwise. In the second case, we show that $ζ_{g,ρ}(0)$ is given by the Reidemeister--Turaev torsion, thus extending Fried's conjecture to a generic set of acyclic (but not necessarily unitary) representations. Our proof relies on computing the dimensions of the spaces of generalized twisted Pollicott--Ruelle resonant states at zero for any $ρ\in U_g$.