Geometric Group Theory and Surface Dynamics

几何群论和表面动力学

• 批准号: 9803638
• 负责人: Michael Handel
• 金额: $ 8.6万
• 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: 1998
• 资助国家: 美国
• 起止时间: 1998-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803638&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Surface; Dynamics

The project concerns pseudo-Anosov surface dynamics and the outer automorphism group of a free group. One part of the project is to analyze a certain partial order on conjugacy classes in the mapping class group of a four times punctured disk. The goal is to find a description of this partial order, in terms of the known combinatorial structure of the mapping class group, that is more transparent and more amenable to computation than the standard dynamical one. The second part of the project extends the principal investigator's joint work with Mark Feighn on coherence of groups. The focus is on deciding which one-relator groups are coherent. The third and final part of the project is a continuation of work with Mladen Bestvina and Mark Feighn on the outer automorphism group of a free group. The goal of this part of the proposal is to formulate and prove a structure theorem for a class of finitely generated subgroups that contains, for example, all abelian subgroups. The pseudo-Anosov homeomorphism is one of the most important concepts in the areas of mathematics known as three-dimensional geometry, three-dimensional topology, and two-dimensional dynamical systems. The study of pseudo-Anosov homeomorphisms began in the mid-1970's. During the decade that followed, a great deal of progress was made in analyzing their behavior. Since then progress has slowed, in part because only the more difficult problems remain. Chief among these problems is an understanding of the "forcing partial order," which, roughly speaking, describes the way in which simple two dimensional systems change into chaotic ones. The principal investigator will use extensive computer calculations and techniques from other areas of mathematics, especially that of geometric group theory to investigate this problem. The application of techniques derived from a variety of mathematical subfields should prove effective in illuminating this problem. The other parts of the project focus on the algebraic analogs of the p seudo-Anosov homeomorphisms and will use techniques that are common to dynamical systems and geometric group theory.

该项目涉及伪阿诺索夫表面动力学和自由群的外自同构群。 该项目的一部分是分析四次穿孔圆盘的映射类群中共轭类的某个偏序。 目标是根据映射类组的已知组合结构找到该偏序的描述，该描述比标准动态结构更透明且更易于计算。 该项目的第二部分扩展了首席研究员与马克·费恩在群体一致性方面的联合工作。 重点在于确定哪些单相关群体是一致的。 该项目的第三部分也是最后一部分是与 Mladen Bestvina 和 Mark Feighn 在自由群的外自同构群上的工作的延续。 该提案这一部分的目标是制定并证明一类有限生成子群的结构定理，该子群包含例如所有阿贝尔子群。 伪阿诺索夫同胚是三维几何、三维拓扑和二维动力系统等数学领域最重要的概念之一。 伪阿诺索夫同胚的研究始于 20 世纪 70 年代中期。 在接下来的十年里，在分析他们的行为方面取得了很大进展。 从那时起，进展已经放缓，部分原因是只剩下更困难的问题了。 这些问题中最主要的是对“强制偏序”的理解，粗略地说，它描述了简单的二维系统转变为混沌系统的方式。 首席研究员将使用广泛的计算机计算和其他数学领域的技术，特别是几何群论的技术来研究这个问题。 来自各种数学子领域的技术的应用应该可以有效地解决这个问题。 该项目的其他部分重点关注伪阿诺索夫同胚的代数类比，并将使用动力系统和几何群论中常见的技术。