Andrew Putman
We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g \times 2g$ symplectic group for $L$ odd and $g \geq 3$.
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For some $g \geq 3$, let $Γ$ be a finite index subgroup of the mapping class group of a genus $g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $Γ$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve $x$. For some $n \geq 1$, we have $T_x^n \in Γ$. We prove that $T_x^n$ is torsion in the abelianization of $Γ$. Our second result shows that the abelianization of $Γ$ is finite if $Γ$ contains a "large chunk" (in a certain…
We determine the abelianization of the symmetric mapping class group of a double unbranched cover using the Riemann theta constant, Schottky theta constant, and the theta multiplier. We also give lower bounds of the abelianizations of some finite index subgroups of the mapping class group.
In this paper, we determine the abelianization of the level d mapping class group for d=2 and odd d. We also extend the homomorphism of the Torelli group defined by Heap to a homomorphism of the level 2 mapping class group.
The rational abelianization of the Chillingworth subgroup of the mapping class group of a surface
本文研究了紧致定向曲面的映射类群的Chillingworth子群的有理化abelization，通过第一Johnson同态和Casson-Morita同态确定其结构，并计算了与该子群相关的中心扩张的Euler类阶数及Casson-Morita同态的核。
We show that the closure of the compactly supported mapping class group of an infinite type surface is not perfect and that its abelianization contains a direct summand isomorphic to an uncountable direct sum of rationals. We also extend this to the Torelli group and show that in the case of surfaces with infinite genus the abelianization of the Torelli group contains an indivisible copy of an uncountable free abelian group as well. Finally we give an application to the question of automatic continuity by exhibiting discontinuous homomorphisms to the rationals.
Abelianization of Symmetric Mapping Class Groups
研究关于闭可定向曲面无分支正则$p$重循环覆盖相关的两个自然群，计算奇素数$p$时它们的交换化，发现与$p = 2$时不同的分裂现象，差异体现在普里姆表示像或约翰逊核特殊元素中。
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 $g\ge4$.
Invariants of $\mathbb{Z}/p$-Homology 3-Spheres from the Abelianization of the Level-p Mapping Class Group
本文研究了定向$\mathbb{Z}/d$-同调3-球面与level-$d$映射类群的关系，提出了一种基于平凡2-上闭链族构造不变量的工具，并借此求出了素数$p$下由level-$p$映射类群阿贝尔化构造的所有不变量，推翻了B. Perron的相关猜想。
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 study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.
Finite generation of abelianizations of the genus 3 Johnson kernel and the commutator subgroup of the Torelli group for $\mathrm{Out}(F_3)$
本文证明了当g=3时，约翰逊核和极小群的交换化是有限生成的，提出了一个新的充分条件来判断模是否有限生成。
Abelianizations of finite-index subgroups of the handlebody group
研究亏格≥4时柄体映射类群是否有非平凡有理交换化有限指数子群这一问题，通过证明子午线多重扭转在特定子群的\(H_1(\Gamma;\mathbb{Q})\)中消失及某些子群的\(H_1(\Gamma;\mathbb{Q}) = 0\)，提供不存在此类子群的证据。
Suppose that $f$ is a homomorphism from the mapping class group $\mathcal{M}(N_{g,n})$ of a nonorientable surface of genus $g$ with $n$ boundary components, to $\mathrm{GL}(m,\mathbb{C})$. We prove that if $g\ge 5$, $n\le 1$ and $m\le g-2$, then $f$ factors through the abelianization of $\mathcal{M}(N_{g,n})$, which is $\mathbb{Z}_2\times\mathbb{Z}_2$ for $g\in\{5,6\}$ and $\mathbb{Z}_2$ for $g\ge 7$. If $g\ge 7$, $n=0$ and $m=g-1$, then either $f$ has finite image (of order at most two if $g\ne 8$), or it is conjugate to one of four "homological representations". As an application we prove th…
Three perfect mapping class groups
本研究证明从2-球面、平面或闭2-圆盘内部移除Cantor集所得曲面的映射类群无真可数指数子群，借助Mann建立的自动连续性完成证明，得到群无真有限指数子群且阿贝尔化平凡的推论。
A toric braid is a braid whose closure is a torus link in $\mathbb{R}^3$. Manturov generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization.
We determine the abelianizations of the following three kinds of graded Lie algebras in certain stable ranges: derivations of the free associative algebra, derivations of the free Lie algebra and symplectic derivations of the free associative algebra. In each case, we consider both the whole derivation Lie algebra and its ideal consisting of derivations with positive degrees. As an application of the last case, and by making use of a theorem of Kontsevich, we obtain a new proof of the vanishing theorem of Harer concerning the top rational cohomology group of the mapping class group with respec…
Let T(N) be the subgroup of the mapping class group of a nonorientable surface N (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for T(N). As an application we compute the first homology group (abelianization) of T(N).
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…
Let A denote either the automorphism group of the free group of rank n>=4 or the mapping class group of an orientable surface of genus n>=12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of A, respectively. For a natural number N denote by G_N the Nth term of the lower central series of G. We prove that (i) any subgroup of G containing [G,G] (in particular, the Johnson kernel in the mapping class group case) is finitely generated; (ii) if N=2 or n>=8N-4 and K is any subgroup of G containing G_N (for instance, K can be the Nth te…
Garoufalidis and Levine introduced the homology cobordism group of homology cylinders over a surface. This group can be regarded as a generalization of the mapping class group. Using torsion invariants, we show that the abelianization of this group is infinitely generated provided that the first Betti number of the surface is positive. In particular, this shows that the group is not perfect. This answers questions of Garoufalidis-Levine and Goda-Sakasai. Furthermore we show that the abelianization of the group has infinite rank for the case that the surface has more than one boundary component…
According to seminal work of Kontsevich, the unstable homology of the mapping class group of a surface can be computed via the homology of a certain lie algebra. In a recent paper, S. Morita analyzed the abelianization of this lie algebra, thereby constructing a series of candidates for unstable classes in the homology of the mapping class group. In the current paper, we show that these cycles are all nontrivial, representing degree 4k+1 homology classes in the homology of the mapping class group of a genus one surface with 4k+1 punctures.
We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Johnson's calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.
The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collecti…
We focus on two kinds of infinite index subgroups of the mapping class group of a surface associated with a Lagrangian submodule of the first homology of a surface. These subgroups, called Lagrangian mapping class groups, are known to play important roles in the interaction between the mapping class group and finite-type invariants of 3-manifolds. In this paper, we discuss these groups from a group (co)homological point of view. The results include the determination of their abelianizations, lower bounds of the second homology and remarks on the (co)homology of higher degrees. As a by-product …
For $4 \nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and…
Higher rank confining subsets and hyperbolic actions of solvable groups
本文扩展了限制子集机制，建立了高秩阿贝尔化可解群的双曲作用与限制子集的对应关系，并据此完整描述了广义可解 Baumslag-Solitar 群的双曲作用，重证了其 BNS 不变量。
Mapping class group of manifolds which look like $3$-dimensional complete intersections
本文计算了形如3维完全交的闭单连通6维流形的映射类群，确定了其阿贝尔化、中心、剩余有限性等代数性质，并给出了生成元及在π_3(M)上平凡作用的子群的生成元与关系。
Mapping class groups of simply connected closed spin 5-manifolds with no 2- and 3-torsion in homology
该研究确定了同调中无2-挠和3-挠的单连通闭自旋5维流形的Torelli群与配边群的同构关系，明确了#^g(S²×S³)的映射类群及其相关代数性质，还研究了S³在S²×S³中的嵌入。
Representation stability for the first homology of congruence subgroups
研究映射类群和自由群自同构群的同余子群的第一同调，以及Torelli群交换化的余不变模在有限域上的模表示序列，确定其合成因子和重数，并建立Church-Farb意义下的周期表征稳定性，应用于研究带level 2结构的曲线模空间上的平坦线丛。
On the non-triviality of the torsion subgroup of the abelianized Johnson kernel
本文通过二维方法证明了abelized Johnson核的torsion子群非平凡，并给出了其基数的下界。
Quotients of the mapping class group by power subgroups
本文研究映射类群关于 Dehn 扭转幂子群的商群，发现其包含无限指标正规子群或 Kähler 子群，并与有限指标子群的阿贝尔化问题等价。
We generalize the notion of a Magnus expansion of a free group in order to extend each of the Johnson homomorphisms defined on a decreasing filtration of the Torelli group for a surface with one boundary component to the whole of the automorphism group of a free group $\operatorname{Aut}(F_{n})$. The extended ones are {\it not} homomorphisms, but satisfy an infinite sequence of coboundary relations, so that we call them {\it the Johnson maps}. In this paper we confine ourselves to studying the first and the second relations, which have cohomological consequences about the group $\operatorname{…
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…
This paper presents fifteen problems about mapping class groups. It is an expanded and updated version of the author's preprint "Ten problems on the mapping class groups". The paper will appear in the book "Problems on Mapping Class Groups and Related Topics", ed. by B. Farb, Proc. Symp. Pure Math. series, Amer. Math. Soc.
This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups.
The mapping class group of a non-exceptional oriented surface of finite type admits a biautomatic structure.
We show that the mapping class group of a compact orientable surface with higher complexity has the following extreme rigidity in the sense of measure equivalence: if the mapping class group is measure equivalent to a discrete group, then they are commensurable up to finite kernel. Moreover, we describe all lattice embeddings of the mapping class group into a locally compact second countable group. We also obtain similar results for finite direct products of mapping class groups.
We show that the mapping class group of an orientable finite type surface has uniformly exponential growth, as well as various closely related groups. This provides further evidence that mapping class groups may be linear.
We prove that the mapping class group of a closed oriented surface of genus $ρ\ge 3$ has no proper subgroup of index $\le 4 ρ+4$.
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。
We investigate the separability of several well known classes of subgroups of the mapping class group of a surface.
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.
This paper presents a number of problems about mapping class groups and moduli space. The paper will appear in the book "Problems on Mapping Class Groups and Related Topics", ed. by B. Farb, Proc. Symp. Pure Math. series, Amer. Math. Soc.
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.
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。
S. Bigelow proved that the braid groups are linear. That is, there is a faithful representation of the braid group into the general linear group of some field. Using this, we deduce from previously known results that the mapping class group of a sphere with punctures and hyperelliptic mapping class groups are linear. In particular, the mapping class group of a closed orientable surface of genus 2 is linear.
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。
The balanced superelliptic mapping class group is the normalizer of the transformation group of the balanced superelliptic covering space in the mapping class group of the total surface. We give finite presentations for the balanced superelliptic mapping class groups of closed surfaces, surfaces with one marked point, and surfaces with one boundary component. To give these presentations, we construct finite presentations for corresponding liftable mapping class groups in a different generating set from Ghaswala-Winarski's presentation in \cite{Ghaswala-Winarski1}.
We prove that the mapping class group of a closed oriented surface of genus at least two does not have Kazhdan's property (T).
Reducible normal generators for mapping class groups are abundant
该研究探讨映射类群的正规生成问题，给出映射类成为正规生成元的条件，提出基于Teichmüller空间渐近平移长度的可约映射类正规生成判别准则，类比了Lanier-Margalit的伪Anosov生成元相关研究。
Group-theoretic Johnson classes and a non-hyperelliptic curve with torsion Ceresa class
本文通过群论方法构造Johnson/Morita余数，应用于曲线的pro-l étale基本群，讨论其与Hain和Matsumoto工作的关系，并给出一个非双曲曲线的Ceresa类在l-adic Abel-Jacobi映射下具有torsion图像的例子。
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.
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.
We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.
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.
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 the mapping class group of a closed connected orientable surface of genus $g$ is generated by three involutions for $g\geq 6$.
We define a generalization of Coxeter graphs and an associated Coxeter system and Coxeter mapping class. These can be used to construct periodic Coxeter mapping classes on surfaces with arbitrarily large genus, preserving lots of symmetries. The periodic mapping classes can in turn be used to construct sequences of pseudo-Anosov mapping classes with bounded normalized dilatation and arbitrarily high genus. We show that the smallest known accumulation point of normalized dilatations can be realized by such a sequence.
We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.
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