arXiv math/00102672000-10-27math.GTmath.GR

On the linearity of certain mapping class groups

Mustafa Korkmaz

备注：: 5 pages, 1 figure, to apper in Turkish Journal of Mathematics

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Linearity of some low-complexity mapping class groups

By analyzing known presentations of the pure mapping groups of orientable surfaces of genus $g$ with $b$ boundary components and $n$ punctures, we show that these groups are isomorphic to some groups related to the braid groups and the Artin group of type $D_4$ in the cases when $g=0$ with $b$ and $n$ arbitrary, and when $g=1$ and $b+n$ is at most $3$. As a corollary, we conclude that the pure mapping class groups are linear in these cases.

2019-03-29math.GRmath.GT

On the linearity problem for mapping class groups

Formanek and Procesi have demonstrated that Aut(F_n) is not linear for n >2. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group, in Aut(F_n), from which they build an FP-group. We first prove that poison groups cannot be embedded in certain mapping class groups. We then show that no FP-groups of any form can be embedded in mapping class groups. Thus the methods of Formanek and Procesi fail in the case of mapping class groups, providing strong evidence that mapping clas…

2001-09-07math.GRmath.GT

On visualization of the linearity problem for mapping class groups of surfaces

We derive two types of linearity conditions for mapping class groups of orientable surfaces: one for once-punctured surface, and the other for closed surface, respectively. For the once-punctured case, the condition is described in terms of the action of the mapping class group on the deformation space of linear representations of the fundamental group of the corresponding closed surface. For the closed case, the condition is described in terms of the vector space generated by the isotopy classes of essential simple closed curves on the corresponding surface. The latter condition also describe…

2014-03-05math.GT

Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification

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.

2010-06-17math.GT

On the linearity of the holomorph group of a free group on two generators

Let F_n denote the free group generated by n letters. The purpose of this article is to show that Hol(F_2), the holomorph of the free group on two generators, is linear. Consequently, any split group extension of F_2 by a linear group H is linear. This result gives a large linear subgroup of Aut(F_3). A second application is that the mapping class group for genus one surfaces with two punctures is linear.

2009-05-03math.GRmath.GT

矩阵元素、单流形和汇合的线性性

Matrix entries, unipotents, and linearity of amalgams

Mathematisches Institut der Universität Bonn（波恩大学数学研究所）；Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克科学促进学会数理生物学研究所）

研究代数群子群汇合的线性性，发现某些线性群的'双'具有线性性，同时获得新的有限生成残余有限群非线性例子。

2026-08-03math.GRmath.GT

更多相关论文收起更多论文

Problems, Questions, and Conjectures about Mapping Class Groups

We discuss a number of open problems about mapping class groups of surfaces. In particular, we discuss problems related to linearity, congruence subgroups, cohomology, pseudo-Anosov stretch factors, Torelli subgroups, and normal subgroups.

2018-06-22math.GT

Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plan…

2009-06-15math.GRmath.GT

On certain questions of the free group automorphisms theory

Certain subgroups of the groups $Aut(F_n)$ of automorphisms of a free group $F_n$ are considered. Comparing Alexander polynomials of two poly-free groups $Cb_4^+$ and $P_4$ we prove that these groups are not isomorphic, despite the fact that they have a lot of common properties. This answers the question of Cohen-Pakianathan-Vershinin-Wu from \cite{CVW}. The questions of linearity of subgroups of $Aut(F_n)$ are considered. As an application of the properties of poison groups in the sense of Formanek and Procesi, we show that the groups of the type $Aut(G*\mathbb Z)$ for certain groups $G$ and …

2007-01-16math.GRmath.GT

Conjugacy classes of finite subgroups of certain mapping class groups

We give a complete description of conjugacy classes of finite subgroups of the mapping class group of the sphere with r marked points. As a corollary we obtain a description of conjugacy classes of maximal finite subgroups of the hyperelliptic mapping class group. In particular, we prove that for a fixed genus g there are at most five such classes.

2005-10-11math.GT

Extending Johnson's and Morita's homomorphisms to the mapping class group

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping class group. From this crossed homomorphism we also obtain a crossed homomorphism extending the $k$th Johnson homomorphism $τ_k$ to the mapping class group. D. Johnson and S. Morita obtained their respective homomorphisms by considering the action of the mapping class group on the ni…

2007-10-01math.GRmath.GT

由某些不可约周期映射类诱导的Teichmüller等距

Teichmüller isometries induced by certain irreducible periodic mapping classes

Indian Institute of Science Education and Research Bhopal（印度科学教育研究所博帕尔分校）

本文提出了一种算法，用于在Fenchel-Nielsen坐标下确定由某些不可约周期映射类诱导的Teichmüller空间等距，并以4g+2阶周期映射类为例给出了描述和几何性质。

2026-06-12math.GT

Heegaard Floer homology of certain mapping tori

We calculate the Heegaard Floer homologies$HF^+(M,s) for mapping tori M associated to certain surface diffeomorphisms, where s is any Spin^c structure on M whose first Chern class is non-torsion. Let gamma and delta be a pair of geometrically dual nonseparating curves on a genus g Riemann surface Sigma_g, and let sigma be a curve separating Sigma_g into components of genus 1 and g-1. Write t-gamma, t_delta, and t_sigma for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms t_gamma^m circ t_delta^n for m,n in Z and that …

2004-09-13math.GTmath.SG

Genericity of pseudo-Anosov mapping classes, when seen as mapping classes

We prove that pseudo-Anosov mapping classes are generic with respect to certain notions of genericity reflecting that we are dealing with mapping classes.

2021-03-02math.GT

运动群与映射类群的拓扑表示——一种统一的函子构造

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。

2024-01-25math.ATmath.GT

Centroids and the Rapid Decay property in mapping class groups

We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).

2010-07-09math.FAmath.GRmath.GT

Representations of the mapping class group of a non-orientable surface

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.

2014-02-19math.GT

Positive factorizations of symmetric mapping classes

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.

2017-10-17math.GT

Finiteness of mapping degree sets for 3-manifolds

By constructing certain maps, this note completes the answer of the Question: For which closed orientable 3-manifold $N$, the set of mapping degrees $cD(M,N)$ is finite for any closed orientable 3-manifold $M$?

2010-10-08math.GT

Integral Riemann-Roch formulae for cyclic subgroups of mapping class groups

The first author conjectured certain relations for Morita-Mumford classes and Newton classes in the integral cohomology of mapping class groups (integral Riemann-Roch formulae). In this paper, the conjecture is verified for cyclic subgroups of mapping class groups.

2006-12-14math.ATmath.GT

每个映射类中曲面辛同胚不动点的精确界

A sharp bound on fixed points of surface symplectomorphisms in each mapping class

本文研究了曲面辛同胚不动点数量问题，通过 Floer 同调计算与扭曲系数方法，得到了每个映射类中不动点的精确下界，将 Poincaré-Birkhoff 定理推广至任意曲面和映射类。

2010-09-03math.DSmath.GTmath.SG