The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister–Turaev torsion

Léo Bénard; Jan Frahm; Polyxeni Spilioti

doi:10.5802/jep.247

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister–Turaev torsion
[Fonction zêta de Ruelle pour les orbifolds hyperboliques de dimension

2

et torsion de Reidemeister-Turaev]

Léo Bénard ¹ ; Jan Frahm ² ; Polyxeni Spilioti ³

¹ Institut de Mathématiques de Marseille, Aix–Marseille Université Site de Saint Charles, 3 place Victor Hugo, Case 19, 13331 Marseille Cedex 3, France
² Department of Mathematics, Aarhus University Ny Munkegade 118, 8000 Aarhus C, Denmark
³ Mathematisches Institut, Georg–August Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany

Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1391-1439.

Résumé
Abstract

Soit $X$ un orbifold hyperbolique de dimension 2, et $X_{1}$ son fibré unitaire tangent. Étant donnée une représentation $ρ : π_{1} (X_{1}) \to GL (V_{ρ})$ , nous étudions dans cet article une fonction zêta dynamique introduite par Ruelle, notée $R (s, ρ)$ , associée à la paire $(X_{1}, ρ)$ . Nous montrons que sa valeur en $s = 0$ est un invariant topologique, la torsion de Reidemeister-Turaev $tor (X_{1}, ρ)$ , si la représentation $ρ$ ne factorise pas par $π_{1} (X)$ . Cela généralise des résultats de Fried, qui avait prouvé $|tor (X_{1}, ρ)| = |R (0, ρ)|$ pour $ρ$ unitaire et classique. Nous levons donc les restrictions sur le choix de $ρ$ , et les indéterminations de phase pour la torsion. Pour cela, nous utilisons la structure d’Euler associée au flot géodésique sur $X_{1}$ . Quand la représentation $ρ$ est un relevé d’une représentation de $π_{1} (X)$ , nous déterminons son ordre d’annulation en $s = 0$ , ainsi que son coefficient dominant.

Let $X$ be a compact hyperbolic surface with finite order singularities, $X_{1}$ its unit tangent bundle. We consider the Ruelle zeta function $R (s; ρ)$ associated to a representation $ρ : π_{1} (X_{1}) \to GL (V_{ρ})$ . If $ρ$ does not factor through $π_{1} (X)$ , we show that the value at $0$ of the Ruelle zeta function equals the sign-refined Reidemeister–Turaev torsion of $(X_{1}, ρ)$ with respect to the Euler structure induced by the geodesic flow and to the natural homology orientation of $X_{1}$ . It generalizes Fried’s conjecture to non-unitary representations, and solves the phase and sign ambiguity in the unitary case. We also compute the vanishing order and the leading coefficient of the Ruelle zeta function at $s = 0$ when $ρ$ factors through $π_{1} (X)$ .

Reçu le : 2022-03-28
Accepté le : 2023-10-10
Publié le : 2023-11-16

DOI : 10.5802/jep.247

Classification : 11M36, 11F72, 37C30, 57Q10
Keywords: Hyperbolic orbisurface, twisted Ruelle zeta function, non-unitary representation, Reidemeister–Turaev torsion, Selberg trace formula
Mot clés : Orbifold hyperbolique, fonctions zêta de Ruelle, torsion de Reidemeister-Turaev, formule des traces de Selberg

Affiliations des auteurs :

Léo Bénard ¹ ; Jan Frahm ² ; Polyxeni Spilioti ³

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {L\'eo B\'enard and Jan Frahm and Polyxeni Spilioti},
     title = {The twisted {Ruelle} zeta function on compact~hyperbolic orbisurfaces and {Reidemeister{\textendash}Turaev} torsion},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
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     year = {2023},
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}

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Léo Bénard; Jan Frahm; Polyxeni Spilioti. The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister–Turaev torsion. Journal de l’École polytechnique — Mathématiques, Tome 10 (2023), pp. 1391-1439. doi : 10.5802/jep.247. https://jep.centre-mersenne.org/articles/10.5802/jep.247/

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