On Fried's conjecture for compact hyperbolic manifolds

Abstract

Fried's conjecture is concerned with the behavior of dynamical zeta functions at the origin. For compact hyperbolic manifolds, Fried proved that for an orthogonal acyclic representation of the fundamental group, the twisted Ruelle zeta function is holomorphic at $s=0$ and its value at $s=0$ equals the Ray-Singer analytic torsion. He also established a more general result for orthogonal representations, which are not acyclic. The purpose of the present paper is to extend Fried's result to arbitrary finite dimensional representations of the fundamental group. The Ray-Singer analytic torsion is replaced by the complex-valued torsion introduced by Cappell and Miller.