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

Abstract

To a transitive pseudo-Anosov flow $\phi$ on a $3$-manifold $M$ and a representation $\rho$ of ${\pi }_1(M)$, we associate a zeta function ${\zeta }_{\phi ,\rho }(s)$ defined for $\Re s\gg 1$, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that ${\zeta }_{\phi ,\rho }(s)$ 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 $\rho$, relates ${\zeta }_{\phi ,\rho }(0)$ 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.

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