$L^2$-torsion invariants and the Magnus representation of the mapping class group

Cohomological Aspects of Magnus Expansions

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{…

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.

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 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 Magnus representation and homology cobordism groups of homology cylinders

A homology cylinder over a compact manifold is a homology cobordism between two copies of the manifold together with a boundary parametrization. We study abelian quotients of the homology cobordism group of homology cylinders. For homology cylinders over general surfaces, it was shown by Cha, Friedl and Kim that their homology cobordism groups have infinitely generated abelian quotient groups by using Reidemeister torsion invariants. In this paper, we first investigate their abelian quotients again by using another invariant called the Magnus representation. After that, we apply the machinery …

关于映射类群的挠生成元

On the Torsion Generators of the Mapping Class Groups

本文研究闭可定向曲面映射类群及扩展映射类群的挠生成元，证明在足够大的亏格下二者均可由两个指定阶数的挠元素生成。

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.

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 $k$.

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.

A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let $\mathbb{F}$ be a field and let $G\subset \mathbb{F}\setminus \{0\}$ be a multiplicative subgroup. We consider the category $\mathcal{Cob}_G$ of $3$-dimensional cobordisms equipped with a representation of their fundamental group in $G$, and the category ${Vect}_{\mathbb{F},\pm G}$ of $\mathbb{F}$-linear maps defined up to multiplication by an element of $\pm G$. Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from $Cob_G$ to ${Vect}_{\mathbb{F},\pm G}$. In particular, when the group $G$ is free abelian and $\mathbb{F}$ is the field of fractions …

同调中无2-挠和3-挠的单连通闭自旋5维流形的映射类群

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。

Torsion Elements in the Mapping Class Group of a Surface

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 $k$.

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.

Stable specific torsion length and periodic mapping classes

We show that for any periodic mapping class, there is some power which maps a nonseparating, simple closed curve to a distinct, disjoint nonseparating curve. As an application of this result, we introduce the notion of stable specific torsion length of a group element and show that the stable specific torsion length of a Dehn twist is bounded above by six.

Characterisation of a class of equations with solutions over torsion-free groups

We study equations over torsion-free groups in terms of their `t-shape' (the occurences of the variable t in the equation). A t-shape is good if any equation with that shape has a solution. It is an outstanding conjecture that all t-shapes are good. In [Klyachko's methods and the solution of equations over torsion-free groups, l'Enseign. Maths. 42 (1996) 49--74] we proved the conjecture for a large class of t-shapes called amenable. In [Tesselations of S^2 and equations over torsion-free groups, Proc. Edinburgh Maths. Soc. 38 (1995) 485--493] Clifford and Goldstein characterised a class of goo…

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。

The Extended Mapping Class Group Can Be Generated by Two Torsions

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.

Homological eigenvalues of mapping classes and torsion homology growth for fibered 3--manifolds

Let S be an orientable surface with negative Euler characteristic, let ψ\in\Mod(S) be a mapping class of S, and let T_ψ be the mapping torus of ψ. We study the action of lifts of ψon the homology of finite covers of S via the torsion homology growth of towers of finite covers of T_ψ. We show that ψadmits a lift to a finite cover with a homological eigenvalue of length greater than one if and only if the mapping torus T_ψ admits a finite cover X and a certain tower of abelian covers which have exponential torsion homology growth. We show that the existence of such a lift of ψis intrinsic to T_ψ…

Ubiquity of geometric finiteness in mapping class groups of Haken 3-manifolds

Mapping class groups of Haken 3-manifolds enjoy many of the homological finiteness properties of mapping class groups of 2-manifolds of finite type. For example, H(M) has a torsionfree subgroup of finite index, which is geometrically finite (i. e. is the fundamental group of a finite aspherical complex). This was proven by J. Harer for 2-manifolds and by the second author for Haken 3-manifolds. In this paper we prove that H(M) acts properly discontinuously on a contractible simplicial complex, with compact quotient. This implies that every torsionfree subgroup of finite index in H(M) is geomet…

A note on geometric constructions of bi-invariant orderings

We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric construction provides a connection between bi-orderings and the rational homotopy theory.

The action of mapping classes on nilpotent covers of surfaces

Let $Σ$ be a surface whose interior admits a hyperbolic structure of finite volume. In this paper, we show that any infinite order mapping class acts with infinite order on the homology of some universal $k$--step nilpotent cover of $Σ$. We show that a Torelli mapping class either acts with infinite order on the homology of a finite abelian cover, or the suspension of the mapping class is a 3--manifold whose fundamental group has positive homology gradient. In the latter case, it follows that the suspended 3--manifold has a large fundamental group. It follows that every element of the Magnus k…

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.

Torsion functions on moduli spaces in view of the cluster algebra

We introduce non-acyclic $PGL_n(\mathbb{C})$-torsion of a 3-manifold with toroidal boundary as an extension of J. Porti's $PGL_2(\mathbb{C})$-torsion, and present an explicit formula of the $PGL_n(\mathbb{C})$-torsion of a mapping torus for a surface with punctures, by using the higher Teichmüler theory due to V. Fock and A. Goncharov. Our formula gives a concrete rational function which represents the torsion function and comes from a concrete cluster transformation associated with the mapping class.

The Existence of Quasimeromorphic Mappings

We prove that a Kleinian group $G$ acting upon $\mathbb{H}^{n}$ admits a non-constant $G$-automorphic function, even if it has torsion elements, provided that the orders of the elliptic (torsion) elements are uniformly bounded. This is accomplished by developing a technique for mashing distinct fat triangulations while preserving fatness.

Abelian subgroups of the mapping class groups for non-orientable surfaces

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.

映射类群阈值处的挠元

Torsion at the Threshold for Mapping Class Groups

本文研究带标记点的闭可定向曲面映射类群的离散泛欧拉类幂次$n=g$的情形，证明该幂次对应的类为挠类，其生成的循环子群的阶是$4g(2g+1)(2g-1)$的正整数倍。

Groupoid Extensions of Mapping Class Representations for Bordered Surfaces

The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class group can be identified with the fundamental group of Riemann's moduli space, it is furthermore identified with a subgroup of the fundamental path groupoid upon choosing a basepoint. A combinatorial model for this, the mapping class groupoid, arises from the invariant cell decomposition of Teichmüller space, whose fundamental path groupoid is called the Ptolem…

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.…

Homology of the curve complex and the Steinberg module of the mapping class group

By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.

Commuting conjugates of finite-order mapping classes

Let $\text{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. In this paper, we derive necessary and sufficient conditions for two finite-order mapping classes to have commuting conjugates in $\text{Mod}(S_g)$. As an application of this result, we show that any finite-order mapping class, whose corresponding orbifold is not a sphere, has a conjugate that lifts under any finite-sheeted cover of $S_g$. Furthermore, we show that any torsion element in the centralizer of an irreducible finite order mapping class is of order at most $2$. We also obtain …

Quasi-homomorphisms and stable lengths in mapping class groups

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.

关于abelized Johnson核的torsion子群非平凡性的研究

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。

Failure of separation by quasi-homomorphisms in mapping class groups

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.

关于对称映射类群的同余子群性质

A congruence subgroup property for symmetric mapping class groups

本文研究了对称映射类群的同余子群性质，并证明了在特定条件下，torsion元素在共轭区分上的性质。

Generating Mapping Class Groups by Involutions

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…

Nielsen realization for infinite-type surfaces

Given a finite subgroup G of the mapping class group of a surface S, the Nielsen realization problem asks whether G can be realized as a finite group of homeomorphisms of S. In 1983, Kerckhoff showed that for S a finite-type surface, any finite subgroup G may be realized as a group of isometries of some hyperbolic metric on S. We extend Kerckhoff's result to orientable, infinite-type surfaces. As applications, we classify torsion elements in the mapping class group of the plane minus a Cantor set, and also show that topological groups containing sequences of torsion elements limiting to the id…

多项式增长映射圆柱的同调增长

Homology growth of polynomially growing mapping tori

本文研究多项式增长自同构的映射圆柱的同调增长性质，证明其具有廉价重建性质并显示torsion同调增长在Farber序列下消失。

关于Torelli群同调中的挠

On torsion in the homology of the Torelli group

本文研究Torelli群同调中由分离简单闭曲线的Dehn扭转生成的abelian cycles，证明它们构成Z/2Z-向量空间，且在k=2且g≥4时有限维。

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…

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。

辛群的有限商与映射类群的比较

Finite quotients of symplectic groups vs mapping class groups

本文通过K理论与Weil表示计算辛群有限商的Schur乘子，并对比量子表示，揭示映射类群在第二同调、剩余有限性及Serre性质A2上的差异。

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。

The Farrell-Jones Conjecture for hyperbolic-by-cyclic groups

We prove the Farrell-Jones Conjecture for mapping tori of automorphisms of virtually torsion-free hyperbolic groups. The proof uses recently developed geometric methods for establishing the Farrell-Jones Conjecture by Bartels-Lück-Reich, as well as the structure theory of mapping tori by Dahmani-Krishna.

Strings and the stable cohomology of mapping class groups

Twenty years ago, Mumford initiated the systematic study of the cohomology ring of moduli spaces of Riemann surfaces. Around the same time, Harer proved that the homology of the mapping class groups of oriented surfaces is independent of the genus in low degrees, increasing with the genus. The (co)homology of mapping class groups thus stabelizes. At least rationally, the mapping class groups have the same (co)homology as the corresponding moduli spaces. This prompted Mumford to conjecture that the stable rational cohomology of moduli spaces is generated by certain tautological classes that he …

Farrell cohomology of low genus pure mapping class groups with punctures

In this paper, we calculate the p-torsion of the Farrell cohomology for low genus pure mapping class groups with punctures, where p is an odd prime. Here, `low genus' means g=1,2,3; and `pure mapping class groups with punctures' means the mapping class groups with any number of punctures, where the punctures are not allowed to be permuted. These calculations use our previous results about the periodicity of pure mapping class groups with punctures, as well as other cohomological tools. The low genus cases are interesting because we know that the high genus cases can be reduced to the low genus…

Non-realizability of the Torelli group as area-preserving homeomorphisms

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion-free subgroups of $\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms.

Automorphisms of hyperbolic groups and graphs of groups

Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a free abelian group, and we determine for which groups $\Out(G)$ is infinite. We also show that there are only finitely many conjugacy classes of torsion elements in $\Out(G)$, for $G$ any torsion-free hyperbolic group. More generally, let $Γ$ be a finite graph of groups decomposition of an arbitrary group $G$ such that edge groups $G_e$ are rigid (i.e\. $\Out…

On power subgroups of mapping class groups

In the first part of this paper we prove that the mapping class subgroups generated by the $D$-th powers of Dehn twists (with $D\geq 2$) along a sparse collection of simple closed curves on an orientable surface are right angled Artin groups. The second part is devoted to power quotients, i.e. quotients by the normal subgroup generated by the $D$-th powers of all elements of the mapping class groups. We show first that for infinitely many $D$ the power quotient groups are non-trivial. On the other hand, if $4g+2$ does not divide $D$ then the associated power quotient of the mapping class group…

Faithful actions of automorphisms on the space of orderings of a group

In this article we study the space of left- and bi-invariant orderings on a torsion-free nilpotent group $G$. We will show that generally the set of such orderings is equipped with a faithful action of the automorphism group of $G$. We prove a result which allows us to establish the same conclusion when $G$ is assumed to be merely residually torsion-free nilpotent. In particular, we obtain faithful actions of mapping class groups of surfaces. We will draw connections between the structure of orderings on residually torsion-free nilpotent, hyperbolic groups and their Gromov boundaries, and we s…

On homology torsion growth

We prove new vanishing results on the growth of higher torsion homologies for suitable arithmetic lattices, Artin groups and mapping class groups. The growth is understood along Farber sequences, in particular, along residual chains. For principal congruence subgroups, we also obtain strong asymptotic bounds for the torsion growth. As a central tool, we introduce a quantitative homotopical method called effective rebuilding. This constructs small classifying spaces of finite index subgroups, at the same time controlling the complexity of the homotopy. The method easily applies to free abelian …

Torsion, TQFT, and Seiberg-Witten invariants of 3-manifolds

We prove a conjecture of Hutchings and Lee relating the Seiberg-Witten invariants of a closed 3-manifold X with b_1 > 0 to an invariant that `counts' gradient flow lines--including closed orbits--of a circle-valued Morse function on the manifold. The proof is based on a method described by Donaldson for computing the Seiberg-Witten invariants of 3-manifolds by making use of a `topological quantum field theory,' which makes the calculation completely explicit. We also realize a version of the Seiberg-Witten invariant of X as the intersection number of a pair of totally real submanifolds of a pr…

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。

A note on the abelianizations of finite-index subgroups of the mapping class group

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…

Non semi-simple TQFTs, Reidemeister torsion and Kashaev's invariants

We construct and study a new family of TQFTs based on nilpotent highest weight representations of quantum sl(2) at a root of unity indexed by generic complex numbers. This extends to cobordisms the non-semi-simple invariants defined in (arXiv:1202.3553) including the Kashaev invariant of links. Here the modular category framework does not apply and we use the ``universal construction''. Our TQFT provides a monoidal functor from a category of surfaces and their cobordisms into the category of graded finite dimensional vector spaces and their degree 0-morphisms and depends on the choice of a roo…