Construction of Anosov flows on fibered hyperbolic 3-manifolds

Abstract

We prove that fibered hyperbolic $3$-manifolds carrying transitive Anosov flows are abundant. More precisely, for every $g\geq 2$, there is a finite index subgroup~$\Gamma$ of $ \mathrm{Mod}(S_g)/\mathrm{Tor}(S_g) \simeq \mathrm{Sp}(2g,\mathbb{Z}) $ so that every element of $\Gamma$ has a representative $\varphi \in \operatorname{Mod}(S_g)$ such that the mapping torus $ M_\varphi := S_g \times [0,1]/(x,1) \sim (\varphi(x),0) $ carries a transitive Anosov flow. The manifold $M_\varphi$ is hyperbolic for almost every element of $\Gamma$. This shows in particular that, in the set of all fibered hyperbolic manifolds, the subset made of the manifolds carrying Anosov flows has positive density up to trivial linear monodromy. Moreover, the subgroup $\Gamma$ is defined by an explicit set of generators, and our construction yields many examples of simple fibered hyperbolic manifolds carrying Anosov flows.