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Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups. (English) Zbl 1500.37030

Author’s abstract: We show that the absolute value at zero of the Ruelle zeta function defined by the geodesic flow coincides with the higher-dimensional Reidemeister torsion for the unit tangent bundle over a 2-dimensional hyperbolic orbifold and a non-unitary representation of the fundamental group. Our proof is based on the integral expression of the Ruelle zeta function. This integral expression is derived from the functional equation of the Selberg zeta function for a discrete subgroup with elliptic elements in \(\mathrm{PSL}_2(\mathbb{R})\). We also show that the asymptotic behavior of the higher-dimensional Reidemeister torsion is determined by the contribution of the identity element to the integral expression of the Ruelle zeta function.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc.
37F31 Quasiconformal methods in holomorphic dynamics; quasiconformal dynamics
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
58J52 Determinants and determinant bundles, analytic torsion
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References:

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