Construction of Anosov flows on fibered hyperbolic 3-manifolds

TL;DR

The paper demonstrates that a positive density of fibered hyperbolic 3-manifolds, constructed via mapping tori of certain surface automorphisms, admit transitive Anosov flows, providing explicit examples and a broad class of such manifolds.

Contribution

It constructs a large class of fibered hyperbolic 3-manifolds with transitive Anosov flows using explicit subgroups of the mapping class group.

Findings

Many fibered hyperbolic 3-manifolds carry transitive Anosov flows.

The subset of such manifolds has positive density among all fibered hyperbolic manifolds.

Explicit construction of examples using generators of a subgroup of the mapping class group.

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.