Geometric Group Theory and Surface Dynamics

该项目涉及伪anosov曲面动力学和自由群的外部自同构群。本课题的一部分是分析四次穿孔盘的映射类群中共轭类的一定偏序。我们的目标是根据已知的映射类组的组合结构，找到一种比标准的动态结构更透明、更易于计算的偏序描述。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。重点是决定哪些单相关组是连贯的。项目的第三部分也是最后一部分是与Mladen Bestvina和Mark Feighn在一个自由群的外部自同构群上的工作的延续。提案的这一部分的目标是制定和证明一类有限生成的子群的结构定理，例如，包含所有的阿贝尔子群。伪anosov同胚是三维几何、三维拓扑和二维动力系统等数学领域中最重要的概念之一。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，在分析它们的行为方面取得了很大进展。自那以后，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转变为混沌系统的方式。首席研究员将使用大量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。从各种数学子领域衍生的技术的应用应该证明对阐明这个问题是有效的。项目的其他部分侧重于p - pseudo - anosov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。

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.