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TWISTED RUELLE ZETA FUNCTION ON HYPERBOLIC MANIFOLDS AND COMPLEX-VALUED ANALYTIC TORSION

Part of: Discontinuous groups and automorphic forms Zeta and $L$-functions: analytic theory Lie groups Abstract harmonic analysis Smooth dynamical systems: general theory Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  22 January 2025

Polyxeni Spilioti Open the ORCID record for Polyxeni Spilioti [Opens in a new window]
Polyxeni Spilioti*
Affiliation:
National Technical University of Athens, School of Applied Mathematical and Physical Sciences, Department of Mathematics, Heroon Polytechneiou 9, 15780 Zografou – Athens, Greece
*
xespil@mail.ntua.gr
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Abstract

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold X. The twisted Ruelle zeta function is associated with an acyclic representation $\chi \colon \pi _{1}(X) \rightarrow \operatorname {\mathrm {GL}}_{n}(\mathbb {C})$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in [6], multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi $.

Keywords

twisted Ruelle zeta functiondeterminant formulaCappell–Miller torsionrefined analytic torsionFried’s conjecture

MSC classification

Primary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems
Secondary: 58J52: Determinants and determinant bundles, analytic torsion 22E45: Representations of Lie and linear algebraic groups over real fields 11F72: Spectral theory; Selberg trace formula 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 43A85: Analysis on homogeneous spaces
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 33
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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