The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of minimal pseudo-Anosov dilatations

Abstract

Let $\delta_{g,n}$ be the minimal dilation of pseudo-Anosovs defined on an orientable surface of genus $g$ with $n$ punctures. It is proved by Tsai that for any fixed $g\ge2$, there exists a constant $c_g$ depending on $g$ such that \[ \frac{1}{c_g}\cdot \frac{\log n}{n} \lt \log \delta_{g,n} \lt c_g \cdot \frac{\log n}{n} \qquad \text{for any }n\ge3 \] This means that the logarithm of the minimal dilatation $\log \delta_{g, n}$ is on the order of $\log n/n$. We prove that if $2g + 1$ is relatively prime to $s$ or $s + 1$ for each $0\le s\le g$, then \[ \limsup_{n\to\infty}\frac{n(\log \delta_{g,n})}{\log n}\le 2 \] holds. In particular, if $2g + 1$ is prime, then the above inequality on $\delta_{g,n}$ holds. Our examples of pseudo-Anosovs $\phi$’s which provide the upper bound above have the following property: The mapping torus $M_\phi$ of $\phi$ is a single hyperbolic 3-manifold $N$ called the magic manifold, or the fibration of $M_\phi$ comes from a fibration of $N$ by Dehn filling cusps along the boundary slopes of a fiber.