Zeta functions and the Fried conjecture for smooth pseudo-Anosov flows

Abstract

To a transitive pseudo-Anosov flow $φ$ on a $3$-manifold $M$ and a representation $ρ$ of $π_1(M)$, we associate a zeta function $ζ_{φ,ρ}(s)$ defined for $\Re s \gg 1$, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that $ζ_{φ,ρ}(s)$ has a meromorphic continuation to $\mathbb{C}$. We also prove a version of the Fried conjecture for smooth pseudo-Anosov flows which, under some conditions on $ρ$, relates $ζ_{φ,ρ}(0)$ to the Reidemeister torsion of $M$. Finally we prove a topological analogue of the Dirichlet class number formula. In order to deal with singularities, we use $C^\infty$ versions of the approaches of Rugh and Sanchez--Morgado, based on Markov partitions.