Generating the mapping class group by three involutions

On the involution generators of the mapping class group of a punctured surface

Let Mod(Sigma_{g, p}) denote the mapping class group of a connected orientable surface of genus g with p punctures. For every even integer p \geq 10 and g \geq 14, we prove that Mod(Sigma_{g, p}) can be generated by three involutions. If the number of punctures p is odd and \geq 9, we show that Mod(Sigma_{g, p}) for g \geq 13 can be generated by four involutions. Moreover, we show that for an even integer p \geq 4 and 3 \leq g \geq 6, Mod(Sigma_{g, p}) can be generated by four involutions.

大映射类群的对合生成元

Involution Generators of the Big Mapping Class Group

本文研究具有 n 个端点且亏格累积的无限曲面的大映射类群，证明当 n ≥ 6 时该群可由五个对合拓扑生成，当 n ≥ 3 时可由六个对合拓扑生成。

Generating the Extended Mapping Class Group by Three Involutions

We prove that the extended mapping class group, $\rm Mod^{*}(Σ_{g})$, of a connected orientable surface of genus $g$, can be generated by three involutions for $g\geq 5$. In the presence of punctures, we prove that $\rm Mod^{*}(Σ_{g,p})$ can be generated by three involutions for $g\geq 10$ and $p\geq 6$ (with the exception that for $g\geq 11$, $p$ should be at least $15$).

Mapping class group is generated by three involutions

We prove that the mapping class group of a closed connected orientable surface of genus at least eight is generated by three involutions.

Self-similar surfaces: involutions and perfection

We investigate the problem of when big mapping class groups are generated by involutions. Restricting our attention to the class of self-similar surfaces, which are surfaces with self-similar ends space, as defined by Mann and Rafi, and with 0 or infinite genus, we show that, when the set of maximal ends is infinite, then the mapping class groups of these surfaces are generated by involutions, normally generated by a single involution, and uniformly perfect. In fact, we derive this statement as a corollary of the corresponding statement for the homeomorphism groups of these surfaces. On the ot…

Positive Dehn Twist Expressions for some New Involutions in the Mapping Class Group II

The positive Dehn twist expressions for the generalizations of the new involutions described in math.GT/0404310 are presented. The homeomorphism types of the Lefschetz fibrations that they define are determined for several examples.

Small torsion generating sets for hyperelliptic mapping class groups

We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping class group is generated by three orientation reversing involutions.

Generating the mapping class group of a punctured surface by involutions

Let $Σ_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\rm Mod(Σ_{\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm Mod(Σ_{\textit{g,b}})$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $\rm Mod(Σ_{\textit{g,b}})$. Brendle and Farb [BF] gave an answer in the case of $g\geq 3, b=0$ and $g\geq 4, b=1$, by describing a generating set consisting of 6 involut…

超椭圆对合的根与映射类群中模去中心的辫子群

Roots of hyperelliptic involutions and braid groups modulo their center inside mapping class groups

本文在亏格为nk的闭曲面映射类群中构造了模去中心的辫子群副本，将半扭辫子类与超椭圆对合等同，从而证明超椭圆对合具有无穷多个平方根和立方根，并讨论其共轭类。

Positive Dehn Twist Expressions for some New Involutions in Mapping Class Group

The well-known fact that any genus $g$ symplectic Lefschetz fibration $ X^{4}\to S^{2}$ is given by a word that is equal to the identity element in the mapping class group and each of whose elements is given by a positive Dehn twist, provides an intimate relationship between words in the mapping class group and 4-manifolds that are realized as symplectic Lefschetz fibrations. In this article we provide new words in the mapping class group, hence new symplectic Lefschetz fibrations. We also compute the signatures of those symplectic Lefschetz fibrations.

Every mapping class group is generated by 6 involutions

Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements suffice to generate Mod_{g,0}. We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Mod_{g,b} but also for the order of those elements. In particular, our main result is that 6 involutions (i.…

Generating the mapping class groups by torsions

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the follow results. For $g \geq 4$, $\text{Mod}(S_g)$ can be generated by 4 torsion elements. Three generators are involutions and the forth one is an order 3 element. $\text{Mod}(S_3)$ can be generated by 5 torsion elements. Four generators are involutions and the fifth one is an order 3 element.

The extended mapping class group is generated by three symmetries

We prove that the extended mapping class group is generated by three orientation reversing involutions.

Generators of the Mapping Class Group of a Nonorientable Punctured Surface

Let $\textrm{Mod}(N_{g, p})$ denote the mapping class group of a nonorientable surface of genus $g$ with $p$ punctures. For $g\geq14$, we show that $\textrm{Mod}(N_{g, p})$ can be generated by five elements or by six involutions.

Point pushing, homology, and the hyperelliptic involution

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. We prove a Birman exact sequence for hyperelliptic Torelli groups, and we show that this sequence splits. As a consequence, we show that the hyperelliptic Torelli group is generated by Dehn twists if and only if it is generated by reducible elements. We also give an application to the kernel of the Burau representation.

Torsion normal generators of the mapping class group of a non-orientable surface

We show that the normal closure of any periodic element of the mapping class group of a non-orientable surface whose order is greater than 2 contains the commutator subgroup, which for $g\geq 7$ is equal to the twist subgroup, and provide necessary and sufficient conditions for the normal closures of involutions to contain the twist subgroup. Finally, we provide a criterion for a periodic element to normally generate $\mathcal{M}(N_g)$ and give examples.

Real elements in the mapping class group of $T^2$

We present a complete classification of elements in the mapping class group of the torus which have a representative that can be written as a product of two orientation reversing involutions. Our interest in such decompositions is motivated by features of the monodromy maps of real fibrations. We employ the property that the mapping class group of the torus is identifiable with $SL(2,\Z)$ as well as that the quotient group $PSL(2,\Z)$ is the symmetry group of the {\em Farey tessellation} of the Poincaré disk.

小 torsion 质数生成器用于大映射类群

Small Torsion Topological Generators for Big Mapping Class Groups

本文研究了大映射类群的最小拓扑生成集，特别是由 torsion 元素组成的生成集，证明了对于 n ≥ 16，映射类群可以由四个 involution 生成，而对于洛奇内斯怪物表面（n=1）和雅各布阶梯表面（n=2），可以由三个 involution 生成。此外，对于偶数 n ≥ 8，映射类群可以由四个阶为 n 的 torsion 元素生成，而对于奇数 n ≥ 8，可以由三个阶为 n 的 torsion 元素和一个阶为 n-1 的 torsion 元素生成。

可定向面映射类群的最小生成：可定向情况的综述

Minimal Generation of Mapping Class Groups: A Survey of the Orientable Case

本文综述了可定向面映射类群的最小生成问题，探讨了从经典结果到最新突破的生成集发展，包括通过三个反交换元生成特定面群的最新结果。

Crosscap slides and the level 2 mapping class group of a nonorientable surface

Crosscap slide is a homeomorphism of a nonorientable surface of genus at least 2, which was introduced under the name Y-homeomorphism by Lickorish as an example of an element of the mapping class group which cannot be expressed as a product of Dehn twists. We prove that the subgroup of the mapping class group of a closed nonorientable surface N generated by all crosscap slides is equal to the level 2 subgroup consisting of those mapping classes which act trivially on H_1(N;Z_2). We also prove that this subgroup is generated by involutions.

Generating mapping class groups of nonorientable surfaces with boundary

We obtain simple generating sets for various mapping class groups of a nonorientable surface with punctures and/or boundary. We also compute the abelianizations of these mapping class groups.

del Pezzo 曲面的对合的同痕类

Isotopy classes of involutions of del Pezzo surfaces

本文研究了 del Pezzo 曲面底流形 $M_n$ 的映射类群，证明了对合的分类与结构定理、光滑 Nielsen 实现问题的肯定解，并利用双曲反射群理论对三种源自双有理几何的特殊对合给出了纯拓扑刻画。

曲面上的交错群作用与对称群作用

Alternating and symmetric actions on surfaces

本文研究亏格至少为2的闭曲面的映射类群，给出两个挠元生成有限对称或交错子群的充要条件，刻画对合提升条件，并证明此类子群不含不可约类与超椭圆对合，最后分类了S_10和S_11上的作用。

无限型曲面映射类群的粗有界生成集

Coarsely bounded generating sets for mapping class groups of infinite-type surfaces

本文通过示例探讨了无限型曲面映射类群是否具有粗有界生成集，以阐明Mann和Rafi的相关假设背景。

The Torsion Generating Set Of The Extended Mapping Class Groups In Low Genus Cases

We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.

Generating the mapping class group of a nonorientable surface by crosscap transpositions

A crosscap transposition is an element of the mapping class group of a nonorientable surface represented by a homeomorphism supported on a one-holed Klein bottle and swapping two crosscaps. We prove that the mapping class group of a compact nonorientable surface of genus $g\ge 7$ is generated by conjugates of one crosscap transposition. In the case when the surface is either closed or has one boundary component, we give an explicit set of $g+2$ crosscap transpositions generating the mapping class group.

A finite generating set for the level 2 mapping class group of a nonorientable surface

We obtain a finite set of generators for the level 2 mapping class group of a closed nonorientable surface of genus $g\ge 3$. This set consists of isotopy classes of Lickorish's Y-homeomorphisms also called crosscap slides.

Generating the spin mapping class group by Dehn twists

Let f be a Z/2Z-spin structureon a closed surface S of genus g>3. We determine a generating set of the stabilizer of f in the mapping class group of S consisting of Dehn twists about an explicit collection of 2g+1 curves in S. If g=3 then we also determine a generating set of the stabilizer of an odd Z/2Z-spin structure consisting of Dehn twists about a collection of 6 curves.

Generating mapping class group by two torsion elements

We prove that the mapping class group of a closed connected orientable surface of genus $g$ is generated by two elements of order $g$ for $g\geq 6$. Moreover, for $g\geq 7$ we found a generating set of two elements, of order $g$ and $g'$ which is the least divisor of $g$ such that $g'>2$. Furthermore, we proved that it is generated by two elements of order $g/\gcd (g,k)$ for $g\geq 3k^2+4k+1$ and any positive integer $k$.

On minimal generating sets for the mapping class group of a punctured surface

Let $Σ_{g,p}$ be a oriented connected surface of genus $g$ with $p$ punctures. We denote by $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ the mapping class group and the extended mapping class group of $Σ_{g,p}$, respectively. In this paper, we show that $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ are generated by two element for $g\geq 3$ and $p\geq 0$.

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.

通过两个元素生成非定向曲面 genus $g \geq 13$ 的映射类群

Generating the mapping class group of a nonorientable surface of genus $g \geq 13$ by two elements

研究证明 genus $g \geq 13$ 的非定向曲面映射类群可由两个元素生成，改进了此前 $g \geq 19$ 的结论。

凸集极对合的拓扑洞察

A topological insight into the polar involution of convex sets

本文证明含原点闭凸集在 Attouch-Wets 度量下同胚于 Hilbert 立方体，且极映射等价于标准取反对合，并刻画了具有唯一不动点的对偶映射。

生成球面循环覆盖的可提升映射类群

Generating the liftable mapping class groups of cyclic covers of spheres

本文推导球面有限叶正则分支循环覆盖的可提升映射类群的有限生成集，给出相关算法，确定同构类，恢复两类覆盖的生成集，推进映射类群研究。

Generating the mapping class group of a nonorientable surface by three torsions

We prove that the mapping class group $\mathcal{M}(N_g)$ of a closed nonorientable surface of genus $g$ different than 4 is generated by three torsion elements. Moreover, for every even integer $k\ge 12$ and $g$ of the form $g=pk+2q(k-1)$ or $g=pk+2q(k-1)+1$, where $p,q$ are non-negative integers and $p$ is odd, $\mathcal{M}(N_g)$ is generated by three conjugate elements of order $k$. Analogous results are proved for the subgroup of $\mathcal{M}(N_g)$ generated by Dehn twists.

Torsion Generators of the Twist Subgroup

We showed that the twist subgroup of the mapping class group of a closed connected nonorientable surface of genus $g\geq13$ can be generated by two involutions and an element of order $g$ or $g-1$ depending on whether $g$ is odd or even respectively.

A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group \mathcal{M}_{g,*}

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus >=2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

关于用伪 Anosov 元素生映射类群

On generating mapping class groups by pseudo-Anosov elements

本文证明闭定向曲面的映射类群可由两个伪 Anosov 元素生成，并在高亏格下得到共轭、伸缩率任意大等更强结论，同时推广至超椭圆映射类群。

Factoring in the hyperelliptic Torelli group

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. The authors and Putman proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic elements.

A minimal generating set of the level 2 mapping class group of a non-orientable surface

We construct a minimal generating set of the level 2 mapping class group of a nonorientable surface of genus $g$, and determine its abelianization for $g\ge4$.

关于超椭圆映射类群的注记

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。

Normal generators for mapping class groups are abundant

We provide a simple criterion for an element of the mapping class group of a closed surface to have normal closure equal to the whole mapping class group. We apply this to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator for the mapping class group when the genus is at least 3. We also give many examples of pseudo-Anosov normal generators, answering a question of D. D. Long. In fact we show that every pseudo-Anosov mapping class with stretch factor less than $\sqrt{2}$ is a normal generator. Even more, we give pseudo-Anosov normal g…

Generating the mapping class group by torsion elements of small order

We show that the mapping class group of a closed oriented surface of genus at least three is generated by 3 elements of order 3 and by 4 elements of order 4. Note that the mapping class group cannot be generated by finitely many torsion elements of same order if genus is equal to one or two.

生成不可定向闭曲面的level $2$映射类群的扭转子群

Generating the twist subgroup of the level $2$ mapping class group of a non-orientable closed surface

本文为不可定向闭曲面的映射类群中由Dehn扭转生成且在Z2同调上平凡作用的子群，构造了两个小生成集。

A small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists

We give a small generating set for the twist subgroup of the mapping class group of a non-orientable surface by Dehn twists. The difference between the number of the generators and a lower bound of numbers of generators for the twist subgroup by Dehn twists is one. The lower bounds is obtained from an argument of Hirose [5].

Generating the surface mapping class group by two elements

Wajnryb proved that the mapping class group of an orientable surface is generated by two elements. We prove that one of these generators can be taken as a Dehn twist. We also prove that the extended mapping class group is generated by two elements, again one of which is a Dehn twist. Another result we prove is that the mapping class groups are also generated by two elements of finite order.

非定向面映射类群的最小生成：非定向情况的综述

Minimal Generation of Mapping Class Groups: A Survey of the Nonorientable Case

本文综述了非定向面映射类群的最小生成理论，揭示了生成集的几何与代数关系，并扩展至有孔面，提供显式生成器和证明。

Generating mapping class groups with elements of fixed finite order

We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We additionally prove similar results for some permutation groups, linear groups, and automorphism groups of free groups.

无限图的纯映射类群的生成集与代数性质

Generating Sets and Algebraic Properties of Pure Mapping Class Groups of Infinite Graphs

本文完全分类了具有纯映射类群且允许粗有界生成集的局部有限无限图，并研究了其代数性质，包括半直积分解、第一积分上同调、剩余有限性和Tits二择性，为这些群的刚性研究奠定基础。

Generators for the mapping class group of a nonorientable surface

We show that Szepietowski's system of generators for the mapping class group of a non-orientable surface is a minimal generating set by Dehn twists and $Y$-homemorphisms.

A finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface

We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove that the level 2 twist subgroup is normally generated in the mapping class group by a crosscap pushing map along a non-separating two-sided simple loop for genus $g\geq 5$ and $g=3$. As an application, we calculate the first homology group of the level 2 twist subgroup for genus $…

用两个周期元素生成扩展映射类群

Generating Extended Mapping Class Groups with Two Periodic Elements

本文研究扩展映射类群的二元有限阶生成问题，证明 $n$ 穿孔球面恰在 $n\ e4$ 时可由两个有限阶元素生成，并据此推出亏格 2 曲面的同类结论。

Minimal generating and normally generating sets for the braid and mapping class groups of the disc, the sphere and the projective plane

We consider the (pure) braid groups B_{n}(M) and P_{n}(M), where M is the 2-sphere S^2 or the real projective plane RP^2. We determine the minimal cardinality of (normal) generating sets X of these groups, first when there is no restriction on X, and secondly when X consists of elements of finite order. This improves on results of Berrick and Matthey in the case of S^2, and extends them in the case of RP^2. We begin by recalling the situation for the Artin braid groups. As applications of our results, we answer the corresponding questions for the associated mapping class groups, and we show th…

Linearly Bounded Conjugator Property for Mapping Class Groups

Given two conjugate mapping classes f and g, we produce a conjugating element w such that |w| < K(|f|+|g|), where |.| denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.