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

Bénard, Léo; Frahm, Jan; Spilioti, Polyxeni

Journal de l’École polytechnique — Mathématiques, Tome 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 $R(s;\rho )$ associated to a representation $\rho :{\pi }_1\left(X_1\right)\to \mathrm{GL}\left(V_{\rho }\right)$. If $\rho$ does not factor through ${\pi }_1\left(X\right)$, we show that the value at $0$ of the Ruelle zeta function equals the sign-refined Reidemeister–Turaev torsion of $(X_1,\rho )$ 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 $\rho$ factors through ${\pi }_1\left(X\right)$.

Soit $X$ un orbifold hyperbolique de dimension 2, et $X_1$ son fibré unitaire tangent. Étant donnée une représentation $\rho :{\pi }_1\left(X_1\right)\to GL\left(V_{\rho }\right)$, nous étudions dans cet article une fonction zêta dynamique introduite par Ruelle, notée $R(s,\rho )$, associée à la paire $(X_1,\rho )$. Nous montrons que sa valeur en $s=0$ est un invariant topologique, la torsion de Reidemeister-Turaev $tor(X_1,\rho )$, si la représentation $\rho$ ne factorise pas par ${\pi }_1\left(X\right)$. Cela généralise des résultats de Fried, qui avait prouvé $\left|tor(X_1,\rho )\right|=\left|R(0,\rho )\right|$ pour $\rho$ unitaire et classique. Nous levons donc les restrictions sur le choix de $\rho$, 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 $\rho$ est un relevé d’une représentation de ${\pi }_1\left(X\right)$, nous déterminons son ordre d’annulation en $s=0$, ainsi que son coefficient dominant.

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

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