Geometric Group Theory and Surface Dynamics

几何群论和表面动力学

• 批准号: 0103435
• 负责人: Michael Handel
• 金额: $ 9.36万
• 依托单位: Research Foundation Of The City University Of New York (Lehman)
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-01 至 2004-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0103435&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Surface; Dynamics

AbstractAward: DMS-0103435Principal Investigator: Michael HandelThe proposal divides into four projects in the related fields oftwo dimensional dynamical systems and geometric group theory.The goal of the first, which is a collaboration with Mark Feighn,is to compute, in a transparent, geometric and algorithmic way,the centralizer of an element of the outer automorphismgroup. The second is to continue the principal investigator'sstudy of the forcing order on braid types in the four timespunctured disk. This project has both an experimental andtheoretical part. Computer programs are used to generateexamples and to aid in the formation of conjectures. Onceconjectures are made that cannot be disproved by computer,rigorous proofs will be attempted. The third project is acollaboration with John Franks with the long term goal of provingthat generic area preserving diffeomorphisms of the twodimensional sphere have dense periodic orbits. The moreimmediate goal is to find analogs for generic area preservingdiffeomorphisms of the topological structure that is known toexist for twist maps. The fourth is a collaboration with LeeMosher. The first steps in the project will be to identify andstudy quasi-lines that can play the role in Culler Vogtmann spacethat Teichmuller geodesics play in Teichmuller space.This proposal is concerned with the interface between two areasof mathematics: two dimensional dynamical systems and geometricgroup theory. The former studies the long term behavior ofsystems that evolve over time, while the latter treats algebraicobjects by geometric means. The fields have been intertwined formore than fifty years and have been the focus of a great deal ofresearch in the past twenty. Part of the proposal focuses on twolong standing fundamental questions in two dimensional dynamicalsystems. The first examines how simple systems change intochaotic ones. The second concerns transformations of the spherethat preserve area, and asks whether every piece of the spherecontains at least one point that eventually (as the systemevolves) returns to its original position. There are two groups(in the technical algebraic sense) that are most closely relatedto two dimensional dynamical systems. They are the mapping classgroup and the outer automorphism group. To understand thegeometry of a group one must understand its geodesics; i.e. whatthe shortest paths are between any two points. The geodesics ofthe mapping class group have been well understood for some time.The principal investigator will generalize from what is knownabout the geodesics of the mapping class group to identify andstudy geodesics for the outer automorphism group.

摘要奖项：DMS-0103435 首席研究员：Michael Handel 该提案分为二维动力系统和几何群论相关领域的四个项目。第一个项目是与 Mark Feighn 合作，其目标是以透明的几何形式进行计算算法方式，外自同构群元素的中心化。二是继续课题组主要研究者对四次刺穿盘中辫状类型的强迫顺序的研究。 该项目既有实验部分，也有理论部分。 计算机程序用于生成示例并帮助形成猜想。一旦做出无法被计算机反驳的猜想，就会尝试严格的证明。 第三个项目是与约翰·弗兰克斯的合作，其长期目标是证明二维球体的保留微分同胚的通用区域具有密集的周期轨道。 更直接的目标是找到保留已知存在于扭曲映射的拓扑结构的微分同胚的通用区域的类似物。 第四个是与 LeeMosher 的合作。 该项目的第一步将是识别和研究准线，这些准线可以在 Culler Vogtmann 空间中发挥作用，就像 Teichmuller 测地线在 Teichmuller 空间中发挥的作用一样。该提案涉及两个数学领域之间的接口：二维动力系统和几何群论。 前者研究随时间演化的系统的长期行为，而后者则通过几何手段处理代数对象。 五十多年来，这些领域一直交织在一起，并且在过去二十年中一直是大量研究的焦点。 该提案的一部分重点关注二维动力系统中两个长期存在的基本问题。 第一个研究了简单系统如何转变为混乱系统。 第二个问题涉及保留面积的球体的变换，并询问球体的每一部分是否至少包含一个点，该点最终（随着系统的演化）返回到其原始位置。 有两组（在技术代数意义上）与二维动力系统关系最密切。 它们是映射类组和外自同构组。 要了解群的几何形状，必须了解其测地线；即任意两点之间的最短路径是什么。 映射类群的测地线已经被很好地理解了一段时间。主要研究者将从映射类群的测地线的已知信息中进行推广，以识别和研究外自同构群的测地线。