The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister–Turaev torsion
Journal de l’École polytechnique — Mathématiques, Volume 10 (2023), pp. 1391-1439.

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 )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).

Soit X un orbifold hyperbolique de dimension 2, et X 1 son fibré unitaire tangent. Étant donnée une représentation ρ:π 1 (X 1 )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.

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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
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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {The twisted {Ruelle} zeta function on compact~hyperbolic orbisurfaces and {Reidemeister{\textendash}Turaev} torsion},
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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, Volume 10 (2023), pp. 1391-1439. doi : 10.5802/jep.247. https://jep.centre-mersenne.org/articles/10.5802/jep.247/

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