Geometric Group Theory

几何群论

• 批准号: 0405979
• 负责人: Lee Mosher
• 金额: --
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405979&HistoricalAwards=false
• 关键词: Geometric; Group; Theory

This project will focus on the geometry of two important classes ofgroups: mapping class groups of surfaces; and outer automorphism groupsof free groups. Mapping class groups will be studied using the tools oflarge scale geometry, with the aim of learning about quasi-isometricproperties of mapping class groups and proving quasi-isometric rigidity.Along the way we expect to develop new tools for understandingquasi-isometric properties of relatively hyperbolic groups. Outerautomorphism groups are not as well understood as mapping class groups,and will be studied from a more foundational point of view. The focuswill be to develop analogies with the theory of Teichmuller spaces whichis a very well developed tool for studying mapping class groups, with theparticular aim of developing analogies with the ending laminationconjecture.Geometric group theory is the study of symmetry groups associated tohighly symmetric geometries of infinite extent, such as a crystal latticein 3-dimensional space, whose symmetry group is the set of all motions ofspace which move each point of the lattice to another point of thelattice. The theory of groups was developed in the 19th century forstudying symmetry, and it quickly developed into a deep and richmathematical tool. Groups and geometries go hand-in-hand: every group hasa corresponding geometry, and one can use tools of geometry to learn aboutthe group. This project will focus on two classes of groups which haveassumed importance in the field of topology. Given a surface, such as thesurface of a doughnut with one or more holes, the "mapping class group" ofthat surface describes all the ways in which that surface can be mappedback onto itself. Similarly, the "outer automorphism group of a freegroup" is a group that describes all of the ways in which a finite1-dimensional network or graph can be mapped back onto itself. In bothcases, we will study geometries associated to these groups, with the aimof learning more about the groups themselves. In the case of mappingclass groups our aim is to prove one of the central geometric conjectures,known as quasi-isometric rigidity. In the case of outer automorphismgroups of free groups, where the geometry is less well understood, we willattempt to build new tools and make a deeper study of existing tools, inorder to reach a better understanding of the geometry.

这个项目将集中于两类重要的群的几何：映射类的曲面群和外自同构群的自由群。我们将利用大尺度几何的工具来研究映射类群，以了解映射类群的拟等距性质，并证明拟等距刚性。同时，我们期望开发新的工具来理解相对双曲群的拟等距性质。外自同构群不像映射类群那样被理解，我们将从更基础的角度来研究它。重点是发展与TeichMuller空间理论的类比，这是一个非常成熟的研究映射类群的工具，特别是目的是发展与结束层合猜想的类比。几何群论是研究与无限范围的高度对称几何有关的对称群，例如三维空间中的晶格，其对称群是将晶格的每个点移动到晶格的另一点的空间的所有运动的集合。群论是在19世纪为研究对称性而发展起来的，并迅速发展成为一种深刻而丰富的数学工具。群和几何是齐头并进的：每个群都有对应的几何，人们可以使用几何工具来了解群。这个项目将关注在拓扑学领域具有重要意义的两类群。给定一个曲面，例如具有一个或多个孔的甜甜圈的曲面，该曲面的“映射类组”描述了该曲面可以映射回其自身的所有方式。类似地，“自由群的外自同构群”是描述有限一维网络或图可以映射回其自身的所有方式的群。在这两种情况下，我们都将学习与这些群相关的几何，目的是更多地了解这些群本身。在映射类群的情况下，我们的目的是证明一个中心几何猜想，即所谓的拟等距刚性。在自由群的外自同构群的情况下，我们将尝试建立新的工具并对现有工具进行更深入的研究，以便更好地理解几何。