We introduce a new object, the dynamical torsion, which extends the potentially ill-defined
value at
of the Ruelle zeta function of a contact Anosov flow, twisted by an acyclic
representation of the fundamental group. We show important properties of the
dynamical torsion: it is invariant under deformations among contact Anosov flows, it
is holomorphic in the representation and it has the same logarithmic derivative as
some refined combinatorial torsion of Turaev. This shows that the ratio between this
torsion and the Turaev torsion is locally constant on the space of acyclic
representations.
In particular, for contact Anosov flows path-connected to the geodesic flow of some
hyperbolic manifold among contact Anosov flows, we relate the leading term of the Laurent
expansion of
at the origin, the Reidemeister torsion and the torsions of the finite-dimensional
complexes of the generalized resonant states of both flows for the resonance
. This
extends previous work of Dang, Guillarmou, Rivière and Shen (Invent. Math. 220:2
(2020), 525–579) on the Fried conjecture near geodesic flows of hyperbolic
-manifolds,
to hyperbolic manifolds of any odd dimension.
