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Anosov表面动力学和自由群的外自同构群。 研究了四次穿孔圆盘映射类群中共轭类的偏序问题。 我们的目标是找到一个描述这个偏序，在已知的组合结构的映射类组，这是更透明的，更经得起计算比标准的动态。 该项目的第二部分扩展了首席研究员与Mark Feighn在群体一致性方面的联合工作。 重点是决定哪些单关系员群体是连贯的。 该项目的第三部分也是最后一部分是与Mladen Besteln和Mark Feighn在自由群的外部自同构群上的工作的延续。 这一部分的目标是制定和证明一个结构定理的一类生成的子群，其中包含，例如，所有的阿贝尔子群。 伪Anosov同胚是数学领域中最重要的概念之一，如三维几何，三维拓扑和二维动力系统。 伪Anosov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的进展。 自那时以来，进展缓慢，部分原因是只剩下更困难的问题。 这些问题中最主要的是对“强迫偏序”的理解，粗略地说，它描述了简单的二维系统变成混沌系统的方式。 主要研究者将使用广泛的计算机计算和其他数学领域的技术，特别是几何群论来研究这个问题。 来自各种数学子领域的技术的应用应证明有效地阐明这个问题。 该项目的其他部分侧重于p-Anosov同胚的代数类似物，并将使用动力系统和几何群论中常见的技术。