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

TL;DR

This paper extends zeta function theory to smooth pseudo-Anosov flows on 3-manifolds, proving meromorphic continuation and a Fried conjecture analogue relating zeta functions to Reidemeister torsion.

Contribution

It introduces a generalized zeta function for smooth pseudo-Anosov flows, proves its meromorphic continuation, and establishes a Fried conjecture analogue linking it to Reidemeister torsion.

Findings

Zeta function $\zeta_{\varphi, ho}(s)$ extends meromorphically to $\mathbb{C}$.

Fried conjecture analogue relates $\zeta_{\varphi, ho}(0)$ to Reidemeister torsion.

Established a topological analogue of the Dirichlet class number formula.

Abstract

To a transitive pseudo-Anosov flow $\varphi$ on a $3$-manifold $M$ and a representation $\rho$ of $\pi_1(M)$, we associate a zeta function $\zeta_{\varphi,\rho}(s)$ defined for $\Re s \gg 1$, generalizing the Anosov case. For a class of ``smooth pseudo-Anosov flows'', we prove that $\zeta_{\varphi,\rho}(s)$ 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 $\zeta_{\varphi,\rho}(0)$ 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.