## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 46, Number 3 (2016), 271-287.

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

Eiko Kin and Mitsuhiko Takasawa

#### 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.

#### Article information

**Source**

Hiroshima Math. J., Volume 46, Number 3 (2016), 271-287.

**Dates**

Received: 27 July 2015

Revised: 11 July 2016

First available in Project Euclid: 25 February 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1487991622

**Digital Object Identifier**

doi:10.32917/hmj/1487991622

**Mathematical Reviews number (MathSciNet)**

MR3614298

**Zentralblatt MATH identifier**

1365.57019

**Subjects**

Primary: 57M27: Invariants of knots and 3-manifolds 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces

Secondary: 37B40: Topological entropy

**Keywords**

Mapping class group pseudo-Anosov dilatation entropy fibered 3-manifold magic manifold

#### Citation

Kin, Eiko; Takasawa, Mitsuhiko. The boundary of a fibered face of the magic 3-manifold and the asymptotic behavior of minimal pseudo-Anosov dilatations. Hiroshima Math. J. 46 (2016), no. 3, 271--287. doi:10.32917/hmj/1487991622. https://projecteuclid.org/euclid.hmj/1487991622