Big mapping class groups: an overview
Javier Aramayona, Nicholas G. Vlamis
该研究为一篇综述，梳理了无限拓扑型曲面映射类群领域的最新进展情况。
我们综述了无限拓扑型曲面映射类群的最新研究进展。
We survey recent developments on mapping class groups of surfaces of infinite topological type.
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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…
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.
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 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.
Big mapping class groups acting on homology
该研究聚焦大映射类群对对应曲面第一同调的作用，给出了诱导同调表示的像的精确刻画，推进了相关群作用与同调表示领域的研究。
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…
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。
We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these `big' mapping class groups and shows that each finite-index subgroup has finite outer automorphism group. As a key ingredient, we prove that all simplicial automorphisms between curve complexes of infinite-type orientable surfaces are induced by homeomorphisms.
Multitwists in big mapping class groups
本文证明无限型曲面紧支撑映射类群的闭包不能由多扭转集合生成。
We prove that the first integral cohomology of pure mapping class groups of infinite type genus one surfaces is trivial. For genus zero surfaces we prove that not every homomorphism to $\mathbb{Z}$ factors through a sphere with finitely many punctures. In fact we get an uncountable family of such maps.
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.
In this note, we prove that the compactly supported mapping class group of a surface containing a genus $3$ subsurface has no realization as a subgroup of the homeomorphism group. We also prove that for certain surfaces with order $6$ symmetries, their mapping class groups have no realization as a subgroup of the homeomorphism group. Examples of such surfaces include the plane minus a Cantor set and the sphere minus a Cantor set.
Orderability of big mapping class groups
本文通过有限型子曲面穷竭归纳构造可数稳定 Alexander 系统，重新证明了带边无限型曲面映射类群的左可序性，并刻画了广义理想弧系统诱导左序的等价条件及其与群拓扑的一致性。
Following the work of Rosendal and Mann and Rafi, we try to answer the following question: when is the mapping class group of an infinite-type surface quasi-isometric to a graph whose vertices are curves on that surface? With the assumption of tameness as defined by Mann and Rafi, we describe a necessary and sufficient condition, called translatability, for a geometrically nontrivial big mapping class group to admit such a quasi-isometry. In addition, we show that the mapping class group of the plane minus a Cantor set is quasi-isometric to the loop graph defined by Bavard, which we believe re…
We study two actions of big mapping class groups. The first is an action by isometries on a Gromov-hyperbolic graph. The second is an action by homeomorphisms on a circle in which the vertices of the graph naturally embed. The first two parts of the paper are devoted to the definition of objects and tools needed to introduce these two actions; in particular, we define and prove the existence of equators for infinite type surfaces, we define the hyperbolic graph and the circle needed for the actions, and we describe the Gromov-boundary of the graph using the embedding of its vertices in the cir…
We show that if $M$ is a closed three manifold with a Heegaard splitting with sufficiently big "handlebody distance" then the subgroup of the mapping class group of the Heegaard surface, which extend to both handlebodies is finite. As a corollary, this implies that under the same hypothesis, the mapping class group of $M$ is finite.
The number of ends of big mapping class groups
研究大映射类群的端数，证明当稳定 avenue 表面有至少一个离散类型端时，映射类群为单端。
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 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.
Big mapping class groups and rigidity of the simple circle
本文研究平面去掉康托集后的映射类群在圆上的作用，证明其作用要么平凡，要么半共轭于简单圆上的唯一极小作用。
We study injective homomorphisms between big mapping class groups of infinite-type surfaces. First, we construct (uncountably many) examples of surfaces without boundary whose (pure) mapping class groups are not co-Hopfian; these are the first examples of injective endomorphisms of mapping class groups (of surfaces with empty boundary) that fail to be surjective. We then prove that, subject to some topological conditions on the domain surface, any continuous injective homomorphism between (arbitrary) big mapping class groups that sends Dehn twists to Dehn twists is induced by homeomorphism. Fi…
A big mapping class acting parabolically on the nonseparating curve graph
本文利用精细的曲线图工具，证明了存在无限类型表面关联的曲线图的抛物线等距变换。
We address the question of determining which mapping class groups of infinite-type surfaces admit nonelementary continuous actions on hyperbolic spaces. More precisely, let $Σ$ be a connected, orientable surface of infinite type with tame endspace whose mapping class group is generated by a coarsely bounded subset. We prove that $\mathrm{Map}(Σ)$ admits a continuous nonelementary action on a hyperbolic space if and only if $Σ$ contains a finite-type subsurface which intersects all its homeomorphic translates. When $Σ$ contains such a nondisplaceable subsurface $K$ of finite type, the hyperboli…
On the large scale geometry of big mapping class groups of surfaces with a unique maximal end
本文研究具有唯一极大端的曲面的大映射类群的大尺度几何，给出全局 CB 的完整刻画，证明局部 CB 蕴含 CB 生成，并提供显式判据及一个非驯服曲面的反例。
A new construction of subgroups of big mapping class groups
本文提出了一种新的方法，通过位移映射和多推映射构造大纯映射类群的子群，这些子群不依赖于已知技术，且不通过表面嵌入诱导。
On the homology of big mapping class groups
本文证明单孔康托树曲面的映射类群是无环的，并由此确定单穿孔康托树曲面映射类群的同调，回答了Calegari和Chen的问题，同时将结果推广到一般二叉树曲面。
Compact and finite-type support in the homology of big mapping class groups
本文研究了无限型曲面映射类群同调中是否存在支撑在紧致或有限型子曲面上的非平凡类，利用可移动子曲面和同调稳定性给出了亏格为正时的几乎完整解答及亏格为零时的部分解答。
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。
Property $P_{\text{naive}}$ for big mapping class groups
本文研究了无限类型曲面映射类群的性质 $P_{\text{naive}}$，证明了对于任意有限个非平凡元素，存在无限阶元素使其满足特定的组合结构。
Large scale geometry of big mapping class groups
本文利用Rosendal粗几何框架，对无限型曲面映射类群进行分类，给出了局部粗有界性、粗有界生成集及有界直径的完全分类。
Let S be a surface and let Mod(S,K) be the mapping class group of S permuting a Cantor subset K of S. We prove two structure theorems for normal subgroups of Mod(S,K). (Purity:) if S has finite type, every normal subgroup of Mod(S,K) either contains the kernel of the forgetful map to the mapping class group of S, or it is `pure', i.e. it fixes the Cantor set pointwise. (Inertia:) for any n element subset Q of the Cantor set, there is a forgetful map from the pure subgroup PMod(S,K) of Mod(S,K) to the mapping class group of (S,Q) fixing Q pointwise. If N is a normal subgroup of Mod(S,K) contain…
These notes are the English version of the paper "Hyperbolicité du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in dynamics. We show that the ray graph, which is the analog of the complex of curves for this surface of infinite type, has infinite diameter and is hyperbolic. We use the action of Gamma on this graph to find an explicit non trivial quasimorphism on Gamma and to show that this group has infinite dimensional second bounded cohomology. Finally we give an example of a …
Let $S$ be an orientable, connected surface with infinitely-generated fundamental group. The main theorem states that if the genus of $S$ is finite and at least 4, then the isomorphism type of the pure mapping class group associated to $S$, denoted $\mathrm{PMap}(S)$, detects the homeomorphism type of $S$. As a corollary, every automorphism of $\mathrm{PMap}(S)$ is induced by a homeomorphism, which extends a theorem of Ivanov from the finite-type setting. In the process of proving these results, we show that $\mathrm{PMap}(S)$ is residually finite if and only if $S$ has finite genus, demonstra…
Topological normal generation of big mapping class groups
本文研究了大映射类群的拓扑正常生成问题，证明了在端空间可数时，映射类群拓扑正常生成当且仅当表面唯一自相似，并构造了不可数多个非telescoping但具有拓扑正常生成映射类群的表面例子。
Birman-Hilden theory for big mapping class groups
本文研究了无限类型表面的分岔覆盖映射的Birman-Hilden性质，并展示了非定向无限类型表面的映射类群可作为其定向双覆盖映射类群的子群。
A Bers type classification of big mapping classes
针对无限型曲面的凸双曲结构空间，研究其测地完备子空间的拓扑性质与Teichmüller子空间分解，依据动力学将大映射类分为三类，并应用于曲面映射类群与模群的同构问题。
For every integer l bigger than one, we find elements x and y in the mapping class group of an appropriate orientable surface S, satisfying the Artin relation of length l. That is, xyx... = yxy..., where each side of the equality contains l terms. By direct computations, we first find elements x and y in Mod(S) satisfying Artin relations of every even length bigger than 6, and every odd length bigger than 1. Then using the theory of Artin groups, we give two more alternative ways for finding Artin relations in Mod(S). The first provides Artin relations of every length greater than 3, while the…
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.
In this paper, we study the relationship between the mapping class group of an infinite-type surface and the simultaneous flip graph, a variant of the flip graph for infinite-type surfaces defined by Fossas and Parlier. We show that the extended mapping class group is isomorphic to a proper subgroup of the automorphism group of the flip graph, unlike in the finite-type case. This shows that Ivanov's metaconjecture, which states that any "sufficiently rich" object associated to a finite-type surface has the extended mapping class group as its automorphism group, does not extend to simultaneous …
We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $Σ$ on itself. Our main results are: (1) All conjugacy classes of $MCG(Σ)$ are meager for every $Σ$, (2) $MCG(Σ)$ has a somewhere dense conjugacy class if and only if $Σ$ has at most two maximal ends and no non-displaceable finite-type subsurfaces, (3) $MCG(Σ)$ has a dense conjugacy class if and only if $Σ$ has a unique maximal end and no non-displaceable finite-type subsurfaces. Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Tru…
Let $S$ be an infinite-type surface and $p\in S$. We show that the Thurston-Veech construction for pseudo-Anosov elements, adapted for infinite-type surfaces, produces infinitely many loxodromic elements for the action of $Mod(S;p)$ on the loop graph $L(S;p)$ that do not leave any finite-type subsurface $S'\subset S$ invariant. Moreover, in the language of Bavard-Walker, Thurston-Veech's construction produces loxodromic elements of any weight. As a consequence of Bavard and Walker's work, any subgroup of $Mod(S;p)$ containing two "Thurston-Veech loxodromics" of different weight has an infinite…
For any surface $Σ$ of infinite topological type, we study the Torelli subgroup ${\mathcal I}(Σ)$ of the mapping class group ${\rm MCG}(Σ)$, whose elements are those mapping classes that act trivially on the homology of $Σ$. Our first result asserts that ${\mathcal I}(Σ)$ is topologically generated by the subgroup of ${\rm MCG}(Σ)$ consisting of those elements in the Torelli group which have compact support. In particular, using results of Birman, Powell, and Putman we deduce that ${\mathcal I}(Σ)$ is topologically generated by separating twists and bounding pair maps. Next, we prove the abstr…
Big mapping class groups with uncountable integral homology
本文证明无限型曲面的紧支撑映射类群闭包和 Torelli 群的整同调在每个正次数上不可数，并将该结论推广到末端空间含拓扑可区分点极限点的一大类曲面的完整映射类群，同时构造了亏格2曲面纯映射类群第一同调中的10阶元素。
We prove that the image of the mapping class group by the representations arising in the SU(2)-TQFT is infinite, provided that the genus is bigger than 2 and the level r of the theory is different from 2,3,4,6. In particular the quotient of the mapping class group by the normaizer of the r-th power of a Dehn twist is infinite if the genus is at least 3 and r is bigger than 12.
By considering appropriate finite covering spaces of closed non-orientable surfaces, we construct linear representations of their mapping class group which have finite index image in certain big arithmetic groups.
Small Sets of Topological Generators for Big Mapping Class Groups
本文研究无限型曲面映射类群的拓扑生成元最小个数，证明对于具有无限亏格和n个端点的曲面S(n)，当n≥8时可由3个元素拓扑生成，n≥3时可由4个元素拓扑生成，并给出Loch Ness Monster曲面S(1)和Jacob's Ladder曲面S(2)的生成集。
The non-peripheral curve graph and divergence in big mapping class groups
本文引入了测量表面端复杂度的数值不变量ζ(Σ)，利用非边缘曲线图C_np(Σ)研究映射类群的粗几何特性，证明其连通、无限直径、Gromov超几何及动作有无界轨道，进而得出映射类群的粗秩无限和发散性二次的结论。
Limit sets of mapping class groups
该研究在射影测度叶状结构空间中为映射类群子群引入两类极限点概念，将Teichmüller测地线射线分类，且证明分类可由定向叶状结构的拓扑性质确定。
A Moduli Space of Marked Hyperbolic Structures for Big Surfaces
本文为负欧拉示性数的大曲面构造带标记、完备且 Nielsen 凸的双曲结构模空间，定义拓扑并证明其有限型情形退化为 Teichmüller 空间，同时验证映射类群作用连续。
The Dehn twist coefficient for big and small mapping class groups
本文提出 Dehn 扭转系数（DTC），将分数 Dehn 扭转系数推广至无限型曲面，表征其为满足特定正性条件的唯一齐次拟态射，并发现 DTC 在无限型曲面上的像可覆盖整个实数集，是检测无理旋转行为的有效工具。
We prove that in genus bigger than $2$, the mapping class group action on $\mathrm{Aff}(\mathbb{C})$-characters is ergodic. This implies that almost every representation $π_1 S \longrightarrow \mathrm{Aff}(\mathbb{C})$ is the holonomy of a branched affine structure on $S$, where $S$ is a closed orientable surface of genus $g \geq 2$.
Topological representations of motion groups and mapping class groups -- a unified functorial construction
本文通过函子方法统一构造同调表示，引入定义在包含映射类群和运动群的范畴上的同调表示函子，统一了Lawrence-Bigelow等已知构造并得到新表示。
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。
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。
This is a survey of Magnus representations with particular emphasis on their applications to mapping class groups and monoids (groups) of homology cobordisms of surfaces. In the first half, we begin by recalling the basics of the Fox calculus and overview Magnus representations for automorphism groups of free groups and mapping class groups of surfaces with related topics. In the latter half, we discuss in detail how the theory in the first half extends to homology cobordisms of surfaces and present a number of applications from recent researches.
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…
We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general account of the fundamental relationships (non-degenerate pairings, embeddings, isomorphisms) between the many different flavours of homological representations. Our motivating examples are the Lawrence-Bigelow representations of the braid groups, which are of central importance in the study of the braid groups themselves, as well as their connections with quantu…
Motivated by the work of Birman about the relationship between mapping class groups and braid groups, we discuss the relationship between the orbit braid group and the equivariant mapping class group on the closed surface $M$ with a free and proper group action in this article. Our construction is based on the exact sequence given by the fibration $\mathcal{F}_{0}^{G}M \rightarrow F(M/G,n)$. The conclusion is closely connected with the braid group of the quotient space. Comparing with the situation without the group action, there is a big difference when the quotient space is $\mathbb{T}^{2}$.
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