All finite groups are involved in the Mapping Class Group
Gregor Masbaum, Alan W. Reid
设$Γ_g$表示亏格$g\geq 1$闭可定向曲面的保向映射类群。本文证明了对于固定的$g$，每个有限群都是$Γ_g$的有限指数子群的商.
Let $Γ_g$ denote the orientation-preserving Mapping Class Group of the genus $g\geq 1$ closed orientable surface. In this paper we show that for fixed $g$, every finite group occurs as a quotient of a finite index subgroup of $Γ_g$.
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The hyperelliptic mapping class group has been studied in various contexts within topology and algebraic geometry. What makes this study tractable is that there is a surjective map from the hyperelliptic mapping class group to a mapping class group of a punctured sphere. The more general family of superelliptic mapping class groups does not, in general, surject on to a mapping class group of a punctured sphere, but on to a finite index subgroup. We call this finite index subgroup the liftable mapping class group. In order to initiate the generalization of results on the hyperelliptic mapping c…
We study Question 7.9 in the paper "Monoids in the mapping class group" by Etnyre and Van Horn-Morris; whether a symmetric mapping class admitting a positive factorization is a lift of a quasi-positive braid. We answer affirmatively for mapping classes satisfying certain cyclic conditions.
Action of the mapping class group on a complex of curves and a presentation for the mapping class group of a surface
S. Gervais给出了曲面的映射类群的有限表示（math.GT/9811162）。我们展示这个演示时没有使用Wajnryb的简单演示。我们使用Harvey定义的曲线复合体来代替Hatcher和Thurston的复合体。
Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification
本文研究了映射类群对拓扑曲面有限覆盖的实同调的作用。我们使用映射类的同调表示来构造映射类群的忠实无限维表示。我们表明，这种表示检测每个映射类的Nielsen-Thurston分类。然后，我们讨论了一些例子，发生在理论的辫群和发展一个类似的理论自同构的自由群。我们以一些开放的问题结束。
Generating mapping class groups of nonorientable surfaces with boundary
我们得到了具有穿孔和/或边界的不可定向曲面的各种映射类群的简单生成集。我们还计算了这些映射类群的阿贝尔化。
We obtain representations of the f-based Mapping Class Group of oriented punctured surfaces from an action of mapping classes on Heisenberg homologies of a circle bundle over surface configurations.
Small torsion generating sets for hyperelliptic mapping class groups
证明了超椭圆映射类群和广义超椭圆映射类群都是由两个扭元生成的。计算了由对合生成的超椭圆映射类群的子群的指数，证明了广义超椭圆映射类群是由三个方向反转的对合生成的.
We prove that the mapping class group of a closed connected orientable surface of genus $g$ is generated by three involutions for $g\geq 6$.
On endomorphisms of surface mapping class groups
证明了可定向曲面的映射类群到有限指数子群上的每一个自同态都是自同构。
The abelianization of a symmetric mapping class group
利用Riemann θ常数、Schottky θ常数和θ乘子，确定了双无分支覆盖的对称映射类群的阿贝尔化.给出了映射类群的某些有限指数子群的阿贝尔化的下界。
Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free subgroup of the mapping class groups.
Structure and Topological Generation of Big Mapping Class Groups
本论文系统介绍了大映射类群的结构与拓扑生成，探讨其与经典有限类型情况的区别，并呈现了关于其拓扑和代数结构的基础结果。
We prove that if a normal subgroup of the extended mapping class group of a closed surface has an element of sufficiently small support then its automorphism group and abstract commensurator group are both isomorphic to the extended mapping class group. The proof relies on another theorem we prove, which states that many simplicial complexes associated to a closed surface have automorphism group isomorphic to the extended mapping class group. These results resolve the metaconjecture of N.V. Ivanov, which asserts that any "sufficiently rich" object associated to a surface has automorphism group…
On mapping class groups and their TQFT representations
该论文聚焦映射类群研究，重点综述拓扑量子场论衍生的有限维表示相关内容。
Conjugacy classes of finite subgroups of certain mapping class groups
给出了具有r个标志点的球面的映射类群的有限子群的共轭类的完整刻画。作为推论，我们得到了超椭圆映射类群的极大有限子群的共轭类的一个刻划。特别是，我们证明了，对于一个固定的属g有最多五个这样的类。
The symplectic representation of the mapping class group is unique
从亏格$g\geq 3$的可定向曲面的映射类群到$\GL(2g,\C)$的任何非平凡同态都共轭于标准辛表示。证明了在小于或等于$3g-3$的维数上，映射类群没有忠实的线性表示.
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.
Mapping class groups of non-orientable surfaces for beginners
本文件是主要针对非专家的小型培训班的笔记。目的是为可定向曲面的映射类群理论提供一种初步的方法。
Representations of the mapping class group of a non-orientable surface
通过考虑闭不可可定向曲面的适当的有限覆盖空间，构造了其映射类群在某些大算术群中具有有限指数像的线性表示.
Big Handlebody Distance Implies Finite Mapping Class Group
证明了如果$M$是一个具有Heegaard分裂的闭三流形，且“两体距离”足够大，则Heegaard曲面的映射类群的子群是有限的.作为推论，这意味着在相同的假设下，$M$的映射类群是有限的。
We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.
The extended mapping class group is generated by three symmetries
证明了扩展的映射类群是由三个方向反转对合生成的。
We prove that the mapping class group of a Heegaard splitting with a distance of at least 3 is finite. However, we have constructed a counterexample with a distance of 2 that disproves this assertion. In addition, the fact that the mapping class group of a Heegaard splitting with a distance of at most 1 is infinite, when combined with our results, provides an answer to the question of the finiteness of mapping class groups as viewed from Heegaard distance.
Outer Automorphisms of Mapping Class Groups of Nonorientable Surfaces
设Ng是亏格g >= 5的连通闭不可定向曲面，Mod（Ng）表示Ng的映射类群.证明了若g为奇数，则Mod（Ng）的外自同构群为平凡或Z;若g为偶数，则Mod（Ng）的外自同构群注入到四洞球面的映射类群中.
Using the Galatius--Kupers--Randal-Williams framework of cellular $E_2$-algebras, we prove a secondary stability theorem for mapping class groups of nonorientable surfaces. As a corollary, we obtain a new best known stability range for the homology of the mapping class groups of nonorientable surfaces with respect to adding torus holes.
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。
In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a big mapping class group contains a nonabelian free group. In contrast, we show that no big mapping class group satisfies the strong Tits alternative enjoyed by finite-type mapping class groups. We also give examples of big mapping class groups that fail to satisfy even the classical Tits alternative and give a proof that every countable group appears as a sub…
Integral Riemann-Roch formulae for cyclic subgroups of mapping class groups
第一作者证明了Morita-Mumford类与Newton类在映射类群的积分上同调（积分Riemann-Roch公式）中的某些关系。本文对映射类群的循环子群证明了这个猜想。
Fifteen problems about the mapping class groups
本文提出了关于映射类群的十五个问题。它是作者预印本“映射类群的十个问题“的扩展和更新版本。这篇论文将发表在B. Farb，Proc. Symp.纯数学系列，美国数学学会。
From braid groups to mapping class groups
本文综述了辫群及其相关群的一些性质，这些性质引出了映射类群的一些问题。
Geometry of the mapping class group II: A biautomatic structure
有限型非例外定向曲面的映射类群具有双自动结构。
We prove asymptotic faithfulness for the quantum $\mathrm{Sp}(4)$ mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the $A_n$ family. The methods used are generalized from the proof of asymptotic faithfulness for skein $SU(2)_k$ mapping class group representations. In short, for any noncentral mapping class a comparison vector is found which allows for the mapping class to be detected.
Measure equivalence rigidity of the mapping class group
证明了紧致可定向曲面的映射类群在测度等价意义下具有如下的极端刚性：若映射类群与离散群测度等价，则它们可在有限核上可解.此外，我们描述了所有的格嵌入的映射类群到一个局部紧的第二可数群。对于映射类群的有限直积也得到了类似的结果。
Uniformly exponential growth and mapping class groups of surfaces
我们证明了可定向有限型曲面的映射类群具有一致指数增长性，以及各种密切相关的群。这进一步证明了映射类群可能是线性的。
The abelianization of the level 2 mapping class group
本文确定了d=2且d为奇数的d阶映射类群的阿贝尔化。我们还将Heap定义的Torelli群的同态推广到2级映射类群的同态。
Small index subgroups of the mapping class group
证明了亏格为$ρ\ge 3$的闭定向曲面的映射类群不存在指数为$\le 4 ρ+4$的真子群.
An explicit presentation for asymptotically rigid mapping class groups
本文通过图群的fundamental群商，显式构造了若干渐近刚性映射类群，并基于Labruère和Paris的工作，计算了表面渐近刚性映射类群的显式表示。
Separable subgroups of mapping class groups
本文研究了曲面的映射类群的几类著名子群的可分性。
This article first answers to questions about connectedness of a new family of graphs on unicellular maps. Answering these questions goes through a description of the mapping class group as surgeries on unicellular maps. We also show how unicellular maps encode subgroups of the mapping group and provide filtrations of the mapping class group. These facts add a layer on the ubiquitous character of unicellular maps.
We prove that each Torelli group of an orientable surface with any number of boundary components is at least exponentially distorted in the mapping class group by using Broaddus-Farb-Putman's techniques. Further we show that the distortion of each Torelli group in the level $d$ mapping class group is the same as that of in the mapping class group.
Some problems on mapping class groups and moduli space
本文提出了关于映射类群和模空间的若干问题。这篇论文将发表在B. Farb，Proc. Symp.纯数学系列，美国数学学会。
Even though big mapping class groups are not countably generated, certain big mapping class groups can be generated by a coarsely bounded set and have a well defined quasi-isometry type. We show that the big mapping class group of a stable surface of infinite type with a coarsely bounded generating set that contains an essential shift has infinite asymptotic dimension. This is in contrast with the mapping class groups of surfaces of finite type where the asymptotic dimension is always finite. We also give a topological characterization of essential shifts.
Notes on hyperelliptic mapping class groups
本文系统研究超椭圆映射类群，给出Putman-Wieland猜想在亏格2情形的反例，并将同余子群性质推广到一般超椭圆映射类群，确定其profinite完备化中多重扭转和开子群的中心化子。
The Novikov conjecture for mapping class groups as a corollary of Hamenstadt's theorem
这个简短的说明显示如何诺维科夫猜想映射类群遵循一个定理的加藤和一个结果定理的Hamenstadt。
Given any connected compact orientable surface, a pair of mapping classes are said to be procongruently conjugate if they induce a conjugate pair of outer automophisms on the profinite completion of the fundamental group of the surface. For example, this occurs if they induce conjugate outer automorphisms on every characteristic finite quotient of the fundamental group. In this paper, it is shown that every procongruent conjugacy class of mapping classes, as a subset of the surface mapping class group, is the disjoint union of at most finitely many conjugacy classes of mapping classes. For any…
Big mapping class groups are not extremely amenable
本文利用Kechris-Pestov-Todorčević机制证明，除平凡群情况外，大映射类群及亏格至少为1的曲面的纯映射类群和紧支撑映射类群均非极端顺从。
On the linearity of certain mapping class groups
S. Bigelow证明了辫群是线性的。也就是说，存在辫群到某个域的一般线性群的忠实表示。利用这一点，我们从以前已知的结果，推出的映射类组的一个球的穿刺和超椭圆映射类群是线性的。特别地，亏格为2的闭可定向曲面的映射类群是线性的。
On Nielsen equivalence classes of two-elements generators of mapping class groups
本文研究闭有向曲面映射类群的二元生成元分类，证明亏格不小于八时存在无穷多个 Nielsen 等价类。
Generating the surface mapping class group by two elements
Wajnryb证明了可定向曲面的映射类群是由两个元素生成的。我们证明了这些生成元中的一个可以看作是Dehn扭曲。我们还证明了扩展的映射类群是由两个元素生成的，其中之一是Dehn扭曲。证明了映射类群也是由两个有穷级元生成的。
We give a criterion to prove that some groups are not acylindrically hyperbolic. As an application, we prove that the mapping class group of an infinite type surface is not acylindrically hyperbolic.
In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.
Actions and irreducible representations of the mapping class group
本文证明了映射类群对曲线复形的作用具有不可分解的稳定子。按照一种方法由于伯格和德拉哈普，这一行动导致建设的不可约酉表示的映射类群。
We prove that the mapping class group of a closed connected orientable surface of genus at least eight is generated by three involutions.
Subgroups of the mapping class group from the geometrical viewpoint
我们调查的Kleinian群和子群的映射类群的表面之间的相似性。
Central Extension of Mapping Class Group via Chekhov-Fock Quantization
在Chekhov-Fock量子化中出现的有限型穿孔曲面的映射类群的中心扩张是Meyer类加Euler类的12倍，这与Kashaev量子化中出现的一致.
Property $P_{\text{naive}}$ for non-orientable big Mapping Class Groups
本文提出可定向大映射类群子群满足性质 $P_{\ ext{naive}}$ 的判据，并借助定向二重复盖推广至不可定向无限型曲面映射类群。
We study mapping class groups of infinite type surfaces with isolated punctures and their actions on the loop graphs introduced by Bavard-Walker. We classify all of the mapping classes in these actions which are loxodromic with a WWPD action on the corresponding loop graph. The WWPD property is a weakening of Bestvina-Fujiwara's weak proper discontinuity and is useful for constructing non-trivial quasimorphisms. We use this classification to give a sufficient criterion for subgroups of big mapping class groups to have infinite-dimensional second bounded cohomology and use this criterion to giv…
We give a short and elementary proof of the non-realizability of the mapping class group via homeomorphisms. This was originally established by Markovic, resolving a conjecture of Thurston. With the tools established in this paper, we also obtain some rigidity results for actions of the mapping class group on Euclidean spaces.
Torsion Elements in the Mapping Class Group of a Surface
给定亏格为$g$的闭曲面中的$r$点的有限集，考虑曲面的映射类群中的扭元保持有限集不变。证明了扭元生成映射类群当且仅当对某个整数$k$有$(g, r) \neq (2, 5k+4)$.
Quasi-homomorphisms on mapping class groups
我们改进了映射类群上的拟同态的构造。知道有无界的拟同态是有界的，当限制到特定的子群时，这是有用的，因为从那时起，人们推断出映射类群不是由这些子群有界生成的。本文扩大了这类子群的类。推广的动机是考虑自由群的一阶理论。
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