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~$Γ$ of $ \mathrm{Mod}(S_g)/\mathrm{Tor}(S_g) \simeq \mathrm{Sp}(2g,\mathbb{Z}) $ so that every element of $Γ$ 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 $Γ$. 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 $Γ$ is defined by an explicit set of generators, and our construction yields many examples of simple fibered hyperbolic manifolds carrying Anosov flows.

Figures (26)

• Figure 1: The simple closed curves $a_i,b_j, c_k, d_l$ on $S_g$.
• Figure 2: The torus $N_{\gamma}\simeq\partial\widehat{M}$, the fibers of its projection onto $\gamma$, the canonical meridian $\mu$, a dynamically oriented section and the frame $(\partial_\theta,\partial_\varphi)$.
• Figure 3: The orbits of $\widehat{X}^t$ on the torus $N_{\gamma}$ (in blue). The canonical meridian $\mu$ (in black) and the canonical longitude $\lambda^{s}$ of $N_{\gamma}$ (in green), and a fibration whose fibers are in the homology class $\mu+2\cdot\lambda^{s}$ and transverse to the orbits of $\widehat{X}$ (in tangerine).
• Figure 4: The case $\operatorname{Twist}(W_{\text{loc}}^s(\gamma),S) = -2$ (assuming that the orientation corresponds to the standard orientation of ${\mathbb R}^3$, and that the left of the figure is identified with the right)
• Figure 5: The genus two surface $S$ and the oriented curves $a_\ell,b_\ell,a_r,b_r,c,d$ (the subscripts $\ell$ and $r$ stand for "left" and "right" with respect to $d$).

Theorems & Definitions (87)

• Theorem 1.5
• Remark 1.6
• Theorem 1.8
• Corollary 1.9
• Example 1.10
• Definition 2.3
• Definition 2.5
• Remark 2.7
• Definition 2.8
• Definition 2.9