Some pseudo-Anosov maps on punctured Riemann surfaces represented by multi-twists

带穿孔的不可定向曲面上具有小伸缩因子的伪 Anosov 同胚

Pseudo-Anosov homeomorphisms of punctured non-orientable surfaces with small stretch factor

本文证明了不可定向曲面上伪 Anosov 同胚的最小伸缩因子渐近阶为 $\ rac{1}{g}$，将 Yazdi 的工作推广至不可定向情形，并详述了不可定向 3-流形的 Thurston 纤维面理论。

关于全穿孔伪阿诺索夫映射的归一化伸缩系数的集合

On the set of normalized dilatations of fully-punctured pseudo-Anosov maps

本文改进全穿孔伪阿诺索夫映射环面 veering 三角剖分中四面体数量的上界，并结合计算方法确定归一化伸缩系数集合的最小元素为黄金比例的平方、最小聚点为其四次方。

全穿伪Anosov映射的最小伸缩因子

Minimal stretch factors of orientation-reversing fully-punctured pseudo-Anosov maps

本文研究了反方向全穿伪Anosov映射的最小伸缩因子，证明了其与曲面欧拉特征的关系，并展示了该界限的渐近紧致性。

关于伪阿诺索夫映射的Bonahon--Wong--Yang不变量

On the Bonahon--Wong--Yang invariants of pseudo-Anosov maps

本文猜想并证明伪阿诺索夫映射的BWY不变量等于其映射环面的1-环不变量，解释拓扑不变性，引入后代版本并验证傅里叶关系。

On the minimum dilatation of braids on punctured discs

We find the minimum dilatation of pseudo-Anosov braids on n-punctured discs for 3 <= n <= 8. This covers the results of Song-Ko-Los (n=4) and Ham-Song (n=5). The proof is elementary, and uses the Lefschetz formula.

Ideal Triangulations of Pseudo-Anosov Mapping Tori

We show how to construct an ideal triangulation of a mapping torus of a pseudo-Anosov map punctured along the singular fibers. This gives rise to a new conjugacy invariant of mapping classes, and a new proof of a theorem of Farb-Leininger-Margalit. The approach in this paper is based on ideas of Hamenstadt.

利用正半扭转构造伪阿诺索夫同胚

A construction of pseudo-Anosov homeomorphisms using positive twists

本文提出一种仅用正半扭转构造穿孔球面及高亏格曲面上伪阿诺索夫同胚的方法，可生成拉伸因子具有特定数论性质（如迹域非全实、伽罗瓦共轭在单位圆上）的显式例子。

Entropy vs volume for pseudo-Anosov maps

We will discuss theoretical and experimental results concerning comparison of entropy of pseudo-Anosov maps and volume of their mapping tori. Recent study of Weil-Petersson geometry of the Teichmüller space tells us that they admit linear inequalities for both sides under some bounded geometry condition. We construct a family of pseudo-Anosov maps which violates one side of inequalities under unbounded geometry setting, present an explicit bounding constant for a punctured torus, and provide several observations based on experiments.

A geometric criterion to be pseudo-Anosov

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the fundamental group of a surface S gives rise to a mapping class on the punctured surface, and we show that such a class is pseudo-Anosov if its geodesic representative is "wide" in some hyperbolic 3-manifold homeomorphic to the trivial interval bundle over S.

On representations of certain pseudo-Anosov maps of Riemann surfaces with punctures

Let $S$ be a Riemann surface of type $(p,n)$ with $3p+n>4$ and $n\geq 1$. Let $α_1,α_2\subset S$ be two simple closed geodesics such that $\{α_1, α_2\}$ fills $S$. It was shown by Thurston that most maps obtained through Dehn twists along $α_1$ and $α_2$ are pseudo-Anosov. Let $a$ be a puncture. In this paper, we study the family $\mathcal{F}(S,a)$ of pseudo-Anosov maps on $S$ that projects to the trivial map as $a$ is filled in, and show that there are infinitely many elements in $\mathcal{F}(S,a)$ that cannot be obtained from Dehn twists along two filling geodesics. We further characterize a…

Pseudo-Anosov maps and pairs of filling simple closed geodesics on Riemann surfaces

Let $S$ be a Riemann surface with a puncture $x$. Let $a\subset S$ be a simple closed geodesic. In this paper, we show that for any pseudo-Anosov map $f$ of $S$ that is isotopic to the identity on $S\cup \{x\}$, $(a, f^m(a))$ fills $S$ for $m\geq 3$. We also study the cases of $0<m\leq 2$ and show that if $(a,f^2(a))$ does not fill $S$, then there is only one geodesic $b$ such that $b$ is disjoint from both $a$ and $f^2(a)$. In fact, $b=f(a)$ and $\{a,f(a)\}$ forms the boundary of an $x$-punctured cylinder on $S$. As a consequence, we show that if $a$ and $f(a)$ are not disjoint. Then $(a,f^m(…

伪Anosov辫的最小膨胀

Minimum dilatations of pseudo-Anosov braids

本文确定了足够大n值下n股伪Anosov辫的最小膨胀δn，证实了相关猜想，并基本解决了n穿孔球面上的最小膨胀问题。

The normal closure of a power of a half-twist has infinite index in the mapping class group of a punctured sphere

In this paper we show that the normal closure of the mth power of a half-twist has infinite index in the mapping class group of a punctured sphere. Furthermore, in some cases we prove that the quotient of the mapping class group of the punctured sphere by the normal closure of a power of a half-twist contains a free abelian subgroup. As a corollary we prove that the quotient of the hyperelliptic mapping class group of a surface of genus at least two by the normal closure of the mth power of a Dehn twist has infinite order, and for some integers m the quotient contains a free nonabelian subgrou…

标准嵌入的火车轨道与具有最小扩张因子的伪Anosov映射

Standardly embedded train tracks and pseudo-Anosov maps with minimum expansion factor

本文证明了全穿孔伪Anosov映射的扩张因子下界 $λ(f)^{|χ(S)|} \ge μ^4 \approx 6.85408$，其证明基于对标准嵌入火车轨道及其权重空间上Thurston辛形式的研究。

Single-cylinder square-tiled surfaces and the ubiquity of ratio-optimising pseudo-Anosovs

In every connected component of every stratum of Abelian differentials, we construct square-tiled surfaces with one vertical and one horizontal cylinder. We show that for all but the hyperelliptic components this can be achieved in the minimum number of squares necessary for a square-tiled surface in that stratum. For the hyperelliptic components, we show that the number of squares required is strictly greater and construct surfaces realising these bounds. Using these surfaces, we demonstrate that pseudo-Anosov homeomorphisms optimising the ratio of Teichmüller to curve graph translation lengt…

On Translation Lengths of Pseudo-Anosov Maps on the Curve Graph

We show that a pseudo-Anosov map constructed as a product of the large power of Dehn twists of two filling curves always has a geodesic axis on the curve graph of the surface. We also obtain estimates of the stable translation length of a pseudo-Anosov map, when two filling curves are replaced by multicurves. Three main applications of our theorem are the following: (a) determining which word realizes the minimal translation length on the curve graph within a specific class of words, (b) giving a new class of pseudo-Anosov maps optimizing the ratio of stable translation lengths on the curve gr…

Vanishing of the Euler class in Power subgroups of the punctured mapping class group

In this paper, we use a result of Dahmani to show that the Euler class of some power subgroup (the subgroup normally generated by a fixed power of Dehn twist about a non-separating curve) is trivial inside the mapping class group of once punctured surface.

An Implementation of the Bestvina-Handel Algorithm for Surface Homeomorphisms

Bestvina and Handel have found an effective algorithm that determines whether a given homeomorphism of an orientable, possibly punctured surface is pseudo-Anosov. We present a software package in Java that realizes this algorithm for surfaces with one puncture. Moreover, the package allows the user to define homeomorphisms in terms of Dehn twists, and in the pseudo-Anosov case it generates images of train tracks in the sense of Bestvina-Handel.

Floer Cohomology of Certain pseudo-Anosov Maps

Floer cohomology is computed for certain elements of the mapping class group of a surface $Σ$ of genus $g>1$ which are compositions of positive and negative dehn twists along some loops in $Σ$. The computations cover a certain class of pseudo-Anasov maps.

双曲一次穿孔环面上的扭转数

Twist numbers on hyperbolic once-punctured tori

本文研究双曲一次穿孔环面上简单测地线的相对扭转数，利用双曲几何与 Farey 图证明其图像稠密，并推出扭转数不能连续延拓至测度层状结构空间。

通过 Dynnikov 坐标检测穿孔圆盘映射类群中的自由积

Detecting Free Products in the Mapping Class Group of Punctured Disks via Dynnikov Coordinates

本文证明了穿孔圆盘上定义完全划分的对立曲线的 Dehn 扭转生成自由群或阿贝尔群自由积，并提出基于 Dynnikov 坐标的算法来判定完全划分并揭示自由积的精确结构。

无不动点的伪阿诺索夫同胚、纽结弗洛尔同调与五叶结

Fixed-point-free pseudo-Anosov homeomorphisms, knot Floer homology and the cinquefoil

本文证明亏格二双曲纤维化纽结在非零分数德恩扭转系数下其伪阿诺索夫代表元必有不动点，从而证明纽结弗洛尔同调可检测五叶结，并给出相关同调理论的应用。

punctured 连接和两个 K3 表面的边界 Dehn 折叠在光滑映射类群中是非平凡的

The Boundary Dehn Twist on a Punctured Connected Sum of Two K3 Surfaces is Nontrivial in the Smooth Mapping Class Group

该研究证明了在 punctured 连接和两个 K3 表面的边界 Dehn 折叠在光滑映射类群中是非平凡的，并展示了相关流形的 spin 性质。

On the quantum Teichmüller invariants of fibred cusped 3-manifolds

We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theories and such that their traces define invariants of mapping tori.

Baseilhac-Benedetti与Bonahon-Liu-Wong-Yang不变量之间的关系

A relation between the Baseilhac-Benedetti and the Bonahon-Liu-Wong-Yang invariants

本文研究了Baseilhac-Benedetti不变量、Bonahon-Liu-Wong-Yang不变量及abelian $\mathfrak{gl}_1$-不变量之间的关系，旨在证明其精确联系。

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

This paper concerns the set $\hat{\mathcal{M}}$ 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. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cus…

水平Goodman手术与伪Anosov流的几乎等价性

Horizontal Goodman surgery and almost equivalence of pseudo-Anosov flows

本文推广了Goodman手术操作，证明对传递伪Anosov流进行水平Goodman手术可产生几乎等价的流，并得出相关等价与结构稳定性结果。

边界Dehn扭转通常是交换子

Boundary Dehn twists are often commutators

对于偶复维数的完全交或它们的连通和（或更一般地，某些广泛类别的光滑流形），具体构造了带边穿孔X的微分同胚a,c，其交换子[a,c]表示边界Dehn扭转的光滑映射类（相对边界），表明在Baraglia-Konno、Kronheimer-Mrowka、J. Lin和Tilton的工作中已知在相对边界的光滑映射类群中非平凡的4流形上的边界Dehn扭转在交换化后变为平凡，推广了Y. Lin的结果。

Factoring periodic maps into Dehn twists

Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g \geq 1$. In this paper, we develop various methods for factoring periodic mapping classes into Dehn twists, up to conjugacy. As applications, we develop methods for factoring certain roots of Dehn twists as words in Dehn twists. We will also show the existence of conjugates of periodic maps of order $4g$ and $4g+2$, for $g\geq 2$, whose product is pseudo-Anosov.

Twist Triviality of Canonical Seifert Surfaces

We generalize the idea of unknotting knots to Seifert surfaces. We define an operation called ribbon twist which serves as the equivalent of a crossing change for knots. A Seifert surface is considered untwisted, the equivalent to unknotted, if it is isotopic to a standardly embedded n-fold punctured torus. A Seifert surface is said to be twist trivial if it can be untwisted by ribbon twists. We show that canonical Seifert surfaces are twist trivial.

An analytic family of representations for the mapping class group of punctured surfaces

We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface. The representations depend on a complex number A with |A| <= 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A|<1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from 1, -1, i, -i the representations are finite-dimensional and isomorphic to the "Hom" version of the well-known TQFT quantum representations. The unitary representations in the interval [-1…

映射类群模扭转幂的刚性

Rigidity of mapping class groups mod powers of twists

本文研究带孔球面映射类群模Dehn扭转幂的商，证明其自同构与拟等距刚性，并发展分层双曲空间技术以给出裤子图刚性的新证明。

关于双穿孔环面 skein 代数的双仿射 Hecke 代数的一个注记

A note on double affine Hecke algebra for skein algebra on twice-punctured torus

本文通过 Iwahori--Hecke 算子的 Heegaard 对偶，为双穿孔环面上的 skein 代数构造了 $C^\vee C_1$ 型双仿射 Hecke 代数的推广，并证明其自同构对应 Dehn 扭转，同时给出经典极限的簇代数构造。

The mapping class group of a punctured surface is generated by three elements

Let $Σ_{g,p}$ be a closed oriented surface of genus $g\geq 1$ with $p$ punctures. Let $\rm Mod(Σ_{\textit{g,p}})$ be the mapping class group of $Σ_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\rm Mod({Σ_{\textit{g,p}}})$ is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken as a Dehn twist. For $p\geq 2$, We proved that $\rm Mod(Σ_{\textit{g,p}})$ is generated by three elements.

Random veering triangulations are not geometric

Every pseudo-Anosov mapping class $φ$ defines an associated veering triangulation $τ_φ$ of a punctured mapping torus. We show that generically, $τ_φ$ is not geometric. Here, the word "generic" can be taken either with respect to random walks in mapping class groups or with respect to counting geodesics in moduli space. Tools in the proof include Teichmüller theory, the Ending Lamination Theorem, study of the Thurston norm, and rigorous computation.

Twist left-veering open books, overtwistedness, looseness and virtual looseness

We introduce twist left-veering mapping classes of punctured surfaces. We prove that a twist left-veering open book supports an overtwisted contact structure and determine when the closed braid coming from the punctures is loose or virtually loose.

关于K3曲面边界Dehn扭转的注记

A note on the boundary Dehn twist of $K3$ surfaces

本文针对带孔K3曲面的边界Dehn扭转，证明其在光滑映射类群阿贝尔化后平凡，证明依托Baraglia-Konno的Spin^C族障碍与K3曲面整体Torelli定理。

Singularities of quadratic differentials and extremal Teichmüller mappings defined by Dehn twists

Let $S$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $ω$ be a pseudo-Anosov map of $S$ that is obtained from Dehn twists along two families $\{A,B\}$ of simple closed geodesics that fill $S$. Then $ω$ can be realized as an extremal Teichmüller mapping on a surface of type $(p,n)$ which is also denoted by $S$. Let $ϕ$ be the corresponding holomorphic quadratic differential on $S$. In this paper, we compare the locations of some distinguished points on $S$ in the $ϕ$-flat metric to their locations with respect to the complete hyperbolic metric. More precisely, we show that all possi…

Weil-Petersson translation length and manifolds with many fibered fillings

We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes …

Action of $\mathrm{M}(0,2n)$ on some kernel spaces coming from $\mathrm{SU}(2)$-TQFT

For $n$ an even number, we study representations of the mapping class group of the $n$-punctured sphere arising from $\mathrm{SU}(2)$-TQFT when all punctures are colored by the same integer $N \geq 1$. We prove that the conjecture of Andersen, Masbaum and Ueno holds for the $4$-punctured sphere for all $N \geq 2$. In the case $n \geq 6$ of punctures, we prove it for the pseudo-Anosovs satisfying a homological condition, namely they should act with a non trivial stretching factor on certain eigenspaces of homology of a $\frac{n}{2}$-fold branched cover considered by McMullen. The main idea is t…

Totally geodesic surfaces in twist knot complements

In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher, Miller and Stover. The construction comes from a family of twist knot complements and their dihedral covers. The case $k=1$ arises from the uniqueness of an immersed totally geodesic thrice-punctured sphere, answering a question of Reid. Applying the proof techniques of the main result, we explicitly construct non-elementary maximal Fuchsian subgroups of infi…

纤维化3-流形的伪Anosov单值群的渐近平移长度与正规生成

Asymptotic translation lengths and normal generation for pseudo-Anosov monodromies of fibered 3-manifolds

该研究针对双曲纤维化3-流形，以曲线复形上的渐近平移长度为工具，得到伪Anosov单值群平移长度上界、正规生成性质，并证明曲线复形归一化渐近平移长度一般无法连续延拓到纤维化面。

实有理曲面自同构的映射类

Mapping classes of Real Rational Surface Automorphisms

本文研究实有理曲面上微分同胚族 $F_n$ 的映射类，通过切开不变曲线证明其限制 $\hat F_n$ 为非 Penner 型伪 Anosov 映射，并在 $n=8$ 时给出 Lehmer 数的新几何实现。

曲面映射类群上的扭转余有限拓扑

The twist-cofinite topology on the mapping class group of a surface

该论文在曲面映射类群上定义了一种新拓扑，由此产生一般性概念，并证明伪阿诺索夫映射具有一般性且拉伸因子可任意大，同时由固定亏格Heegaard分解和一般性粘合得到的3-流形一般具有双曲性、零第一贝蒂数和精确的Heegaard亏格。

On groups generated by two positive multi-twists: Teichmueller curves and Lehmer's number

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these determine the same set of Teichmueller curves as the non-obtuse lattice triangles which were classified by Kenyon, Smillie, and Puchta. We also identify a pseudo-Anosov automorphism whose dilatation is Lehmer's number, and show that this is minimal for the groups under considerati…

小曲面的$\rm{PGL}_2$ skein代数

$\rm{PGL}_2$ skein algebras of small surfaces

本文研究小曲面（闭环面与一次穿孔环面）的$\rm{PGL}_2$ skein代数，给出显式有限表示、显式基及其乘法结构，并计算自然映射的核与代数中心，结果依赖于量子参数$q$。

First-return maps of Birkhoff sections of the geodesic flow

This paper compares different pseudo-Anosov maps coming from different Birkhoff sections of a given flow. More precisely, given a hyperbolic surface and a collection of periodic geodesics on it, we study those Birkhoff sections for the geodesic flow on the unit bundle to the surface bounded by the collection. We show that there is a canonical identification of all those surfaces, and that the first-return maps induced by the flow can all be expressed as a composition of negative Dehn twists along the same family of curves: only the order depends on the choice of a particular surface.

带孔曲面的 skein 和聚类代数

Skein and cluster algebras of punctured surfaces

本文通过几何方法证明了带孔曲面的标记 skein 代数与上聚类代数的等价性，并展示了其在代数几何中的应用。

来自纤维化双曲3-流形群的有限商群的Taut多项式

Taut polynomials from finite quotients of fibered hyperbolic 3-manifold groups

本文证明，当单值映射为完全穿孔时，纤维化双曲3-流形群的有限商群能检测Thurston范数球的纤维化面的taut多项式，并发展了扭曲多变量Alexander多项式的profinite不变性的一般框架。