超椭圆平移面上的简单闭测地线及 $\mathcal{H}^{\mathrm{hyp}}(4)$ 中平移面的分类定理

Geodesic length spectrum of hyperelliptic connected components

We propose a general framework for studying pseudo-Anosov homeomorphisms on translation surfaces. This new approach, among other consequences, allows us to compute the systole of the Teichmueller geodesic flow restricted to the hyperelliptic connected components, settling a question of Farb. We stress that all proofs and computations are performed without the help of a computer. As a byproduct, our methods give a way to describe the bottom of the lengths spectrum of the hyperelliptic components.

Bers' constants for punctured spheres and hyperelliptic surfaces

This article is dedicated to prove Buser's conjecture about Bers' constants for spheres with cusps (or marked points) and for hyperelliptic surfaces. More specifically, our main theorem states that any hyperbolic sphere with $n$ cusps has a pants decomposition with all of its geodesics of length bounded by a constant roughly square root of $n$. Other results include lower and upper bounds for Bers' constants for hyperelliptic surfaces and spheres with boundary geodesics.

Existence of closed geodesics through a regular point on translation surfaces

We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also construct explicit examples showing that such points exist. For a surface in any hyperelliptic component, we show that this finite exceptional set is actually empty. The proofs of our results use Apisa's classifications of periodic points and of $\GL(2,\R)$ orbit closures in hyperellip…

Dehn twist presentations of hyperelliptic periodic diffeomorphisms on closed surfaces

We classify up to conjugacy the group generated by a commuting pair of a periodic diffeomorphism and a hyperelliptic involution on an oriented closed surface. This result can be viewed as a refinement of Ishizaka's result on classification of the mapping classes of hyperelliptic periodic diffeomorphisms. As an application, we obtain the Dehn twist presentations of hyperelliptic periodic mapping classes, which are closely related to the ones obtained by Ishizaka.

超椭圆平移曲面中的相对收缩

Relative systoles in hyperelliptic translation surfaces

本文研究了超椭圆平移曲面，证明了面积为一的平移曲面连通分支上的收缩函数存在非全局最大值的局部最大值，当且仅当该连通分支不是超椭圆的。

The hyperelliptic mapping class group of a nonorientable surface of genus $g\geq 4$ has a faithful representation into $GL(g^2-1,\mathbb{R})$

We prove that the hyperelliptic mapping class group of a nonorientable surface of genus $g\geq 4$ has a faithful linear representation of dimension $g^2-1$ over $\mathbb{R}$.