Geometric Group Theory and Surface Dynamics

摘要奖：DMS-0103435主要研究员：Michael Handel该提案分为四个项目，涉及二维动力系统和几何群论的相关领域。第一个项目是与Mark Feighn合作，其目标是以一种透明的、几何的和算法的方式计算外自同构群的元素的中心化。二是继续主要研究者对四次穿孔盘中编织型的强迫顺序的研究。该项目包括实验和理论两个部分。计算机程序被用来生成例子和帮助形成猜想。一旦做出计算机不能反驳的猜想，就会尝试严格的证明。第三个项目是与John Franks的合作，长期目标是证明二维球体的一般面积保持微分同态存在稠密的周期轨道。更直接的目标是为保持已知扭曲图存在的拓扑结构的微分同态的类属区域找到类似的。第四个是与LeeMosher的合作。该项目的第一步将是确定和研究可以在卡勒伏格特曼空间中发挥作用的准线，而TeichMuller测地线在TeichMuller空间中扮演着同样的角色。这项建议涉及到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，而后者用几何方法处理代数对象。这些领域已经交织了50多年，在过去的20年里一直是大量研究的焦点。该提案的一部分集中于二维动力系统中的两个长期存在的基本问题。第一个研究简单的系统如何转变为混沌的系统。第二个问题是关于保存区域的球体的变换，并询问球体的每一块是否至少包含一个最终(随着系统的演化)返回到其原始位置的点。有两类(在技术代数意义上)与二维动力系统关系最密切。它们是映射类群和外自同构群。要了解一个群的几何，就必须了解它的测地线，即任何两点之间的最短路径是什么。映射类群的测地线已被人们熟知了一段时间，主要的研究者将从已知的映射类群测地线的知识中推广出外自同构群的测地线。

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.