Mustafa Korkmaz
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.
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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 $g\ge4$.
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.
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。
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].
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.
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。
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 $…
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。
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…
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.
We prove that the mapping class group of a closed connected orientable surface of genus $g$ is generated by three involutions for $g\geq 6$.
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.
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。
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.
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.
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.
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.
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.
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 $k$.
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 $p\geq 0$.
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…
We prove that, for $g\geq19$ the mapping class group of a nonorientable surface of genus $g$, $\textrm{Mod}(N_g)$, can be generated by two elements, one of which is of order $g$. We also prove that for $g\geq26$, $\textrm{Mod}(N_g)$ can be generated by three involutions if $g\geq26$.
Let $N_{g}$ denote the closed non-orientable surface of genus $g$ and let ${\mathcal M} _g$ denote the mapping class group of $N_{g}$. Let ${\mathcal T} _g$ denote the twist subgroup of ${\mathcal M} _g$ which is the subgroup of ${\mathcal M} _g$ is generated by all Dehn twists. In this thesis, we proved that ${\mathcal T} _g$ is generated by six involutions for $g \geq 16$ or $g = 14$.
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。
In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group.
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.
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。
Let $N_{g,n}$ denote the closed non-orientable surface of genus $g$ with $n$ punctures and let ${\mathcal N}_{g,n}$ denote the mapping class group of $N_{g,n}$. Szepietowski showed that ${\mathcal N}_{g,n}$ is generated by finitely many involutions. The number of elements in his generating set depends linearly on $g$ and $n$. In the case of $n=0$, Szepietowski found an involution generating set in such a way that the number of its elements does not depend on $g$, showing that ${\mathcal N}_{g,0}$ is generated by four involutions. In this thesis, for $n \geq 0$, we prove that ${\mathcal N}_{g,n…
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。
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.
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 $15$).
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…
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.
We give elementary applications of quasi-homomorphisms to growth problems in groups. A particular case concerns the number of torsion elements required to factorise a given element in the mapping class group of a surface.
We give an infinite presentation for the mapping class group of a non-orientable surface. The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops.
Let $Σ_{g,b}$ denote a closed oriented surface genus $g$ with $b$ punctures and let $Mod_{g,b}$ denote its mapping class group. Luo proved that if the genus is at least 3, the group $Mod_{g,b}$ 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 $Mod_{g,b}$. Brendle and Farb gave a partial answer in the case of closed surfaces and surfaces with one puncture, by describing a generating set consisting of 7 involutions. Our main result generalizes the…
We show that mapping class groups of surfaces of genus at least two contain elements of infinite order that are not conjugate to their inverses, but whose powers have bounded torsion lengths. In particular every homogeneous quasi-homomorphism vanishes on such an element, showing that elements of infinite order not conjugate to their inverses cannot be separated by quasi-homomorphisms.
Pseudo-Anosovs are exponentially generic in mapping class groups
本文证明在映射类群中，通过适当选取有限生成集，非伪阿诺索夫映射类在半径球中的比例指数级趋于零，且方法适用于弱双曲群。
The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering in the mapping class group of the total surface. We prove that the balanced superelliptic mapping class groups with either one marked point, one boundary component, or no marked points and boundary are generated by three elements. To prove this, we also show that its liftable mapping class groups are also generated by three elements. These generating sets are minimal except for several no marked points and boundary cases.
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。
Let $S$ be a compact orientable surface, and $\Mod(S)$ its mapping class group. Then there exists a constant $M(S)$, which depends on $S$, with the following property. Suppose $a,b \in \Mod(S)$ are independent (i.e., $[a^n,b^m]\not=1$ for any $n,m \not=0$) pseudo-Anosov elements. Then for any $n,m \ge M$, the subgroup $<a^n,b^m>$ is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in $<a^n,b^m>$ are pseudo-Anosov. We also show that there exists a constant $N$, which depends on $a,b$, such that $<a^n,b^m>$ is free of rank two and convex-…
We prove that the handlebody subgroup of the Torelli group of an orientable surface is generated by genus one BP-maps. As an application, we give a normal generating set for the handlebody subgroup of the level $d$ mapping class group of an orientable surface.
Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements. One of these generators is an order 2 element, and the other one is an order 4g+2 element.
We construct new monomorphisms between mapping class groups of surfaces. The first family of examples injects the mapping class group of a closed surface into that of a different closed surface. The second family of examples are defined on mapping class groups of once-punctured surfaces and have quite curious behaviour. For instance, some pseudo-Anosov elements are mapped to multi-twists. Neither of these two types of phenomena were previously known to be possible although the constructions are elementary.
Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space called the handle graph on which the Torelli group acts cocompactly.
The girth of a finitely generated group G is the supremum of the girth of Cayley graphs for G over all finite generating sets. Let G be a finitely generated subgroup of the mapping class group Mod(S), where S is a compact orientable surface. Then, either G is virtually abelian or it has infinite girth; moreover, if we assume that G is not infinite cyclic, these alternatives are mutually exclusive.
A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group ΠA_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group ΠA_n has close connections with hyperelliptic mapping class groups, braid groups, congruence subgroups of GL(n,Z), and symmetric automorphisms of free groups. We obtain a generating set for the subgroup of ΠA_n consisting of those elements acting trivially on the abelianisation of F_n, the palindromic Torelli gr…
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.
We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.
Positive Dehn twist products for some elements of finite order in the mapping class group of a 2-dimensional closed, compact, oriented surface $Σ_g$, which are rotations of $Σ_g$ through $2π/p$, are presented. The homeomorphism invariants of the resulting simply connected symplectic 4- manifolds are computed.
For $n$ at least 7 and $n$ equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on $n$ strands. These generating sets are of the smallest possible cardinality. For $n$ equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid group on 4 strands cannot be generated by fewer than three elements.
We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of pseudo-Anosov elements to that of all elements in a ball with center at the identity tends to one as the radius of the ball tends to infinity.
Generating the liftable mapping class groups of regular cyclic covers
本文研究了正则循环覆盖的可提升映射类群，证明了其构成无限族自正规化子群（素数叶时为极大子群），并推导出了多种情形下的显式有限生成集。
The balanced superelliptic handlebody group is the normalizer of the transformation group of the balanced superelliptic covering space in the handlebody group of the total space. We prove that the balanced superelliptic mapping class group is generated by four elements. To prove this, we also proved that the liftable Hilden group is generated by three elements. This generating set for the liftable Hilden group is minimal except for some hyperelliptic cases and the generating set for the balanced superelliptic mapping class group above is also minimal for several cases.
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.…
Given a finite set of $r$ points in a closed surface of genus $g$, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if $(g, r) \neq (2, 5k+4)$ for some integer $k$.
The Torelli group, I(S_g), is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface. There are three types of elements that naturally arise in studying I(S_g): bounding pair maps, separating twists, and simply intersecting pair maps (SIP-maps). Historically the first two types of elements have been the focus of the literature on I(S_g), while SIP-maps have received relatively little attention until recently, due to an infinite presentation of I(S_g) introduced by Andrew Putman that uses all three types of elements. We will give a topol…
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.
We prove that the set of non-pseudo-Anosov elements in the Torelli group is exponentially small.
For k >= 1, let Torelli_g^1(k) be the k-th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k, there exists some G_k >= 0 such that Torelli_g^1(k) is generated by elements which are supported on subsurfaces whose genus is at most G_k. We also prove similar theorems for the Johnson filtration of Aut(F_n) and for certain mod-p analogues of the Johnson filtrations of both the mapping class group and of Aut(F_n). The main tools used in the proofs are the related theories of FI-modules (due to the first author together…
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