Mathematical Sciences: Outer Automorphisms and Surface Dynamics

9504912 Handel这个项目涉及自由群的外自同构群和伪Anosov表面动力学。第一部分是继续与Mladen Bestvina和Mark Feighn就三个问题进行联合工作。第一个是多项式增长外自同构子群的刻画。二是证明了外自同构群的所有虚拟可解子群都是虚拟可解子群，从而推广了他们对Tits备选方案的证明。三是证明了外自同构的映射环面是凝聚的。项目的另一部分目标是使用组合群论方法，特别是折叠，分析穿孔圆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，后者可能令人惊讶地为研究前者提供了一种强有力的方法。在动力系统中，重点放在一个不断变化或演化的系统中发生的长期模式上，例如一个围绕行星及其太阳运行的系统。组合群论是用拓扑学和组合学方法研究称为群的代数对象。该项目的一部分研究了一类重要的二维动力系统，称为伪阿诺索夫同胚，并使用组合群论中的一种名为“折叠”的工具来分析这一族中不同成员之间的关系。该项目的另一部分涉及一个群，该群的元素是伪Anosov同胚射的代数类似。虽然这一领域的大多数研究都是从几何学出发的，但在这个项目中，首席研究员将增加动力系统技术，以便更详细地研究这一群体。***

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. ***