arXiv math/01031482001-03math.GTmath.GR

On the linearity problem for mapping class groups

Tara E. Brendle, Hessam Hamidi-Tehrani

备注：: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-23.abs.html
发表信息：: Algebr. Geom. Topol. 1 (2001) 445-468
DOI：: 10.2140/agt.2001.1.445

英文摘要：

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 class groups may in fact be 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

On the linearity of certain mapping class groups

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.

2000-10-27math.GRmath.GT

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 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

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

更多相关论文收起更多论文

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 the 2-linearity of the free group

We construct an action of the free group $F_n$ on the homotopy category of projective modules over a finite dimensional zigzag algebra. The main theorem in the paper is that this action is faithful. We describe the relationship between homotopy classes of paths in the punctured disc and complexes of projective zigzag modules and explore the connection between gradings on the zigzag algebra and monoids in $F_n$. We use this connection to give homological constructions of the standard and Bessis dual word length metrics on the free group.

2016-06-21math.GRmath.GTmath.RT

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

Homological Construction of Quantum Representations of Mapping Class Groups

We provide a homological model for a family of quantum representations of mapping class groups arising from non-semisimple TQFTs (Topological Quantum Field Theories). Our approach gives a new geometric point of view on these representations, and it gathers into one theory two of the most promising constructions for investigating linearity of mapping class groups. More precisely, if $\varSigma_{g,1}$ is a surface of genus $g$ with $1$ boundary component, we consider a (crossed) action of its mapping class group $\mathrm{Mod}(\varSigma_{g,1})$ on the homology of its configuration space $\mathrm{…

2023-03-08math.GTmath.QA