Elmas Irmak, Mustafa Korkmaz
Let S be a compact, connected, orientable surface of positive genus. Let HT(S) be the Hatcher-Thurston complex of S. We prove that Aut(HT(S)) is isomorphic to the extended mapping class group of S modulo its center.
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Let $R$ be a compact, connected, orientable surface of genus $g$ with $n$ boundary components with $g \geq 2$, $n \geq 0$. Let $\mathcal{N}(R)$ be the nonseparating curve graph, $\mathcal{C}(R)$ be the curve graph and $\mathcal{HT}(R)$ be the Hatcher-Thurston graph of $R$. We prove that if $λ: \mathcal{N}(R) \rightarrow\mathcal{N}(R)$ is an edge-preserving map, then $λ$ is induced by a homeomorphism of $R$. We prove that if $θ: \mathcal{C}(R) \rightarrow \mathcal{C}(R)$ is an edge-preserving map, then $θ$ is induced by a homeomorphism of $R$. We prove that if $R$ is closed and $τ: \mathcal{HT}…
Hatcher-Thurston complex for surfaces with non-planar ends
本文提出了一种用于无限类型曲面的复形，该复形在结构上类似于Hatcher-Thurston复形，并证明其自同构群与扩展映射类群同构。
Let $S_{1}$ and $S_{2}$ be connected orientable surfaces of genus $g_{1}, g_{2} \geq 3$, $n_{1},n_{2} \geq 0$ punctures, and empty boundary. Let also $φ: \mathcal{HT}(S_{1}) \rightarrow \mathcal{HT}(S_{2})$ be an edge-preserving alternating map between their Hatcher-Thurston graphs. We prove that $g_{1} \leq g_{2}$ and that there is also a multicurve of cardinality $g_{2} - g_{1}$ contained in every element of the image. We also prove that if $n_{1} = 0$ and $g_{1} = g_{2}$, then the map $\widetildeφ$ obtained by filling the punctures of $S_{2}$, is induced by a homeomorphism of $S_{1}$.
We consider an oriented surface S and a cellular complex X of curves on S, defined by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of S, following the ideas of Hatcher-Thurston and Harer.
We give a characterization of the action of the mapping class group on Thurston's space of measured laminations.
For any atoroidal iwip $ϕ\in Out(F_N)$ the mapping torus group $G_ϕ=F_N\rtimes_ϕ<t>e$ is hyperbolic, and the embedding $ι: F_N \overset{\lhd}{\longrightarrow} G_ϕ$ induces a continuous, $F_N$-equivariant and surjective {\em Cannon-Thurston map} $\hat ι: \partial F_N \to \partial G_ϕ$. We prove that for any $ϕ$ as above, the map $\hat ι$ is finite-to-one and that the preimage of every point of $\partial G_ϕ$ has cardinality $\le 2N$. We also prove that every point $S\in \partial G_ϕ$ with $\ge 3$ preimages in $\partial F_N$ has the form $(wt^m)^\infty$ where $w\in F_N, m\ne 0$, and that there a…
Using a similar algorithm to Hatcher-Thurston's algorithm for finding a presentation of the mapping class group of a surface, Wajnryb succeeded to find a presentation for the handlebody group. This is long and complicated. In this note I simplify Wajnryb's presentation for the handlebody group of genus g = 2.
The free factor complex and the dualizing module for the automorphism group of a free group
本文回答了Hatcher-Vogtmann的问题，证明了对于n=5，自由因子复形的顶同调群并非Aut(F_n)的对偶化模。
We provide analogues for non-orientable surfaces with or without boundary or punctures of several basic theorems in the setting of the Thurston theory of surfaces which were developed so far only in the case of orientable surfaces. Namely, we provide natural analogues for non-orientable surfaces of the Fenchel-Nielsen theorem on the parametrization of the Teichmüller space of the surface, the Dehn-Thurston theorem on the parametrization of measured foliations in the surface, and the Hatcher-Thurston theorem, which gives a complete minimal set of moves between pair of pants decompositions of th…
Using the works of Gervais, Harer, Hatcher and Thurston and others, we show that the mapping class group of a compact orientable surface has a presentation so that the generators are the set of all Dehn twists and the relations are supported in subsurfaces homeomorphic to the one-holed torus or the four-holed sphere. It turns out that all the relations were discovered by Dehn in 1938.
The Thurston norm of 2-bridge link complements
本文研究2桥链补集的Thurston范数单位球，证明其至多有8个面，并借助本质曲面刻画纤维结构及构造顶点数任意多的2分支链。
In this paper, we study a flag complex which is naturally associated to the Thurston theory of surface diffeomorphisms for compact connected orientable surfaces with boundary. The various pieces of the Thurston decomposition of a surface diffeomorphism, thick domains and annular or thin domains, fit into this flag complex, which we call the complex of domains. The main result of this paper is a computation of the group of automorphisms of this complex. Unlike the complex of curves, introduced by Harvey, for which, for all but a finite number of exceptional surfaces, by the works of Ivanov, Kor…
The braided Thompson group $\mathcal B$ is an asymptotic mapping class group of a sphere punctured along the standard Cantor set, endowed with a rigid structure. Inspired from the case of finite type surfaces we consider a Hatcher-Thurston cell complex whose vertices are asymptotically trivial pants decompositions. We prove that the automorphism group $\hat{\mathcal B^{\frac{1}{2}}}$ of this complex is also an asymptotic mapping class group in a weaker sense. Moreover $\hat{\mathcal B^{\frac{1}{2}}}$ is obtained by $\mathcal B$ by first adding new elements called half-twists and further comple…
S. Gervais gave a finite presentation for the mapping class group of a surface (math.GT/9811162). We show this presentation without using Wajnryb's simple presentation. We use a complex of curves defined by Harvey, in place of Hatcher and Thurston's complex.
Cutting and Pasting in the Torelli subgroup of Out($F_n$)
本文借鉴三维流形与曲面映射类群的方法，研究带边界自由群自同构群的Torelli子群，证明其有限生成并满足相应Birman正合序列。
The asymmetry of Thurston's earthquake flow
本文证明 Thurston 地震流在 orbifold 自同构群中正规化子极小，因而不能扩展为 SL(2,R)-作用，也不具有连续重整化自对称，与 Mirzakhani 的可测共轭结果形成对比。
We classify isotopy classes of automorphisms (self-homeomorphisms) of 3-manifolds satisfying the Thurston Geometrization Conjecture. The classification is similar to the classification of automorphisms of surfaces developed by Nielsen and Thurston, except an automorphism of a reducible manifold must first be written as a suitable composition of two automorphisms, each of which fits into our classification. Given an automorphism, the goal is to show, loosely speaking, either that it is periodic, or that it can be decomposed on a surface invariant up to isotopy, or that it has a "dynamically nic…
Thurston's Legacy at the Interface of Surface Dynamics, Train Tracks, and Arithmetic
本章探讨低维动力学、符号动力学与几何拓扑的交叉，围绕拉伸因子的代数刻画与渐近行为，关联组合领域与纤维三维流形，梳理瑟斯顿相关理论的应用。
Dilation surfaces, or twisted quadratic differentials, are variants of translation surfaces. In this paper, we study the question of what elements or subgroups of the mapping class group can be realized as affine automorphisms of dilation surfaces. We show that dilation surfaces can have exotic Dehn twists in their affine automorphism groups and will establish that only certain types of mapping class group elements can arise as affine automorphisms of dilation surfaces. We also generalize a construction of Thurston that constructs a translation surface from a pair of filling multicurves to dil…
The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{ü}ller space with Thurston's Finsler metric
本文研究了负欧拉示性数闭曲面的Teichmüller空间上Thurston度量诱导的切空间和余切空间范数的单位球面的组合结构，给出了切空间单位球面各面维数的公式，证明了该组合结构独立于Teichmüller空间中的基点，并建立了扩展映射类群与单位球面组合自同构群之间的自然同构（亏格2时为满同态），同时得到了余切空间单位球面面的余维数公式以及面暴露和对应投影加权多曲线的充要条件。
Minimal genus Seifert surface for 11 crossing alternating knots
本文计算并明确描述了所有11个交叉点的素交错纽结的Kakimizu复形，结合已知算法与Murasugi和及缝合流形理论，完整刻画了这些纽结的最小亏格Seifert曲面结构。
We study the behavior of hyperbolic affine automorphisms of a translation surface which is infinite in area and genus that is obtained as a limit of surfaces built from regular polygons studied by Veech. We find that hyperbolic affine automorphisms are not recurrent and yet their action restricted to cylinders satisfies a mixing-type formula with polynomial decay. Then we consider the extent to which the action of these hyperbolic affine automorphisms satisfy Thurston's definition of a pseudo-Anosov homeomorphism. In particular we study the action of these automorphisms on simple closed curves…
We study the arc complex of a surface with marked points in the interior and on the boundary. We prove that the isomorphism type of the arc complex determines the topology of the underlying surface, and that in all but a few cases every automorphism is induced by a homeomorphism of the surface. As an application we deduce some rigidity results for the Fomin-Shapiro-Thurston cluster algebra associated to such a surface. Our proofs do not employ any known simplicial rigidity result.
We study the nilpotent part $N'$ of a pseudo-periodic automorphism $h$ of a real oriented surface with boundary $Σ$. We associate a quadratic form $Q$ defined on the first homology group (relative to the boundary) of the surface $Σ$. Using the twist formula and techniques from mapping class group theory, we prove that the form $\tilde{Q}$ obtained after killing ${\ker N}$ is positive definite if all the screw numbers associated with certain orbits of annuli are positive. We also prove that the restriction of $\tilde Q$ to the absolute homology group of $Σ$ is even whenever the quotient of the …
The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ_g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or -1 on the surface depending on whether g=0, 1 or \ge 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface Σ_g where ci…
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