Minimal dilatations of pseudo-Anosovs generated by the magic $3$–manifold and their asymptotic behavior

Abstract

This paper concerns the set $\widehat{\mathcal{ℳ}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic $3$–manifold $N$ by Dehn filling three cusps with a mild restriction. Let $N\left(r\right)$ be the manifold obtained from $N$ by Dehn filling one cusp along the slope $r\in \mathbb{Q}$. We prove that for each $g$ (resp. $g\not\equiv 0\left(mod6\right)$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\widehat{\mathcal{ℳ}}$ defined on a closed surface ${\Sigma }_g$ of genus $g$ is achieved by the monodromy of some ${\Sigma }_g$–bundle over the circle obtained from $N\left(\frac{3}{-2}\right)$ or $N\left(\frac{1}{-2}\right)$ by Dehn filling both cusps. These minimizers are the same ones identified by Hironaka, Aaber and Dunfield, Kin and Takasawa independently. In the case $g\equiv 6\left(mod12\right)$ we find a new family of pseudo-Anosovs defined on ${\Sigma }_g$ with orientable invariant foliations obtained from $N\left(-6\right)$ or $N\left(4\right)$ by Dehn filling both cusps. We prove that if ${\delta }_g^{+}$ is the minimal dilatation of pseudo-Anosovs with orientable invariant foliations defined on ${\Sigma }_g$, then

$$\underset{g\equiv 6\left(mod12\right),g\to \infty }{limsup}glog{\delta }_g^{+}\le 2log\delta \left(D_5\right)\approx 1.0870,$$

where $\delta \left(D_n\right)$ is the minimal dilatation of pseudo-Anosovs on an $n$–punctured disk. We also study monodromies of fibrations on $N\left(1\right)$. We prove that if ${\delta }_{1,n}$ is the minimal dilatation of pseudo-Anosovs on a genus $1$ surface with $n$ punctures, then

$$\underset{n\to \infty }{limsup}nlog{\delta }_{1,n}\le 2log\delta \left(D_4\right)\approx 1.6628.$$