Mathematical Sciences: Outer Automorphisms and Surface Dynamics

数学科学：外自同构和表面动力学

• 批准号: 9504912
• 负责人: Michael Handel
• 金额: $ 8.3万
• 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: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504912&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Outer; Automorphisms; Surface

9504912 Handel The project concerns the outer automorphism group of a free group and pseudo-Anosov surface dynamics. One part is a continuation of joint work with Mladen Bestvina and Mark Feighn concerning three problems. The first is a characterization of subgroups of polynomial growth outer automorphisms. The second is to extend their proof of the Tits Alternative by showing that all virtually solvable subgroups of the outer automorphism group are virtually abelian. The third is to show that the mapping torus of an outer automorphism is coherent. The goal of the other part of the project is to use combinatorial group theory methods, especially folding, to analyze a dynamically defined partial order on conjugacy classes in the mapping class group of a punctured disk. Two fields of mathematics interact in this project: dynamical systems and combinatorial group theory, the latter, perhaps surprisingly, contributing a powerful method for studying the former. In dynamical systems, the focus is on the long-term patterns that occur in a changing, or evolving system, for example a system of orbiting planets and their sun. Combinatorial group theory is the study of algebraic objects called groups using topological and combinatorial methods. One part of the project examines an important family of two-dimensional dynamical systems known as pseudo-Anosov homeomorphisms and uses a tool from combinatorial group theory called "folding" to analyze the relationship between various members of this family. Another part of the project relates to a group whose elements are the algebraic analogs of the pseudo-Anosov homeomorphisms. While most research in this area is derived from geometry, in this project the principal investigator will add dynamical systems techniques in order to investigate this group in greater detail. ***

9504912 Handel 该项目涉及自由群的外自同构群和伪阿诺索夫表面动力学。 其中一部分是与 Mladen Bestvina 和 Mark Feighn 继续就三个问题进行合作。 第一个是多项式增长外自同构子群的表征。第二个是通过证明外自同构群的所有实际上可解的子群实际上都是交换子群来扩展他们对 Tits Alternative 的证明。 第三个是证明外自同构的映射环面是相干的。 该项目的另一部分的目标是使用组合群论方法，特别是折叠，来分析穿孔盘的映射类群中共轭类的动态定义的偏序。 在这个项目中，两个数学领域相互作用：动力系统和组合群论，后者为研究前者提供了一种强大的方法，这或许令人惊讶。 在动力系统中，重点是变化或演化系统中发生的长期模式，例如轨道行星及其太阳的系统。 组合群论是使用拓扑和组合方法研究称为群的代数对象的学科。 该项目的一部分研究了一个重要的二维动力系统家族，称为伪阿诺索夫同胚，并使用组合群论中称为“折叠”的工具来分析该家族各个成员之间的关系。 该项目的另一部分涉及一个群，其元素是伪阿诺索夫同胚的代数类似物。 虽然该领域的大多数研究都源自几何学，但在该项目中，主要研究者将添加动力系统技术，以便更详细地研究该组。 ***