Mathematical Sciences: The Geometry of Groups and Real Trees

数学科学：群和真实树的几何

• 批准号: 9626699
• 负责人: Mark Feighn
• 金额: $ 4.14万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626699&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Geometry; Groups; Real

9626699 Feighn In the current project, the Principal Investigator will continue his work, joint with M. Bestvina and M. Handel, on the structure of the outer automorphism group of a free group. The methods used are a mixture of techniques from E. Rips's theory of real trees and from the train track theory of Bestvina and Handel. The Principal Investigator will also examine to what extent the JSJ theorem of Rips and Z. Sela can be extended to splittings over other than cyclic groups. Now is an exciting time to be a geometric group theorist. Much beautiful mathematics is devoted to classification results. For example, the classification of surfaces (spaces, such as the sphere, that are locally the same as a plane) is classical. This classification is akin to Mendeleev's periodic table of the elements -- each surface appears once and only once. In recent years, due in large part to the geometric ideas of W. Thurston, there has been great progress towards a classification theorem for 3-manifolds (spaces that are locally like 3- dimensional space). Inspired by these manifold successes, mathematicians have begun to look at groups from a geometric viewpoint. (A group is a fundamental object found throughout mathematics and the sciences. Roughly speaking, a group is the set of symmetries of some object.) Mathematics has arrived at the point where the possibility of a classification theorem for (geometric) groups can be envisioned. For example, a recent result of E. Rips and Z. Sela describes how to break a group into more fundamental pieces. In the current project, the Principal Investigator will continue his research along these lines. More specifically, he will show how to break up groups into fundamental pieces in ways other than that of Rips and Sela. Also, he will continue his work with M. Bestvina and M. Handel undertaking a detailed study of some of the groups that will play a key role in this classification. ***

9626699 Feighn在当前项目中，首席研究员将继续与M. Bestvina和M. Handel的联合工作，该组织关于自由组的外部自动形态组的结构。 所使用的方法是E. Rips的真实树理论以及Bestvina和Handel的火车轨道理论的混合物。 首席研究者还将检查撕裂和Z. sela的JSJ定理在多大程度上可以扩展到与循环群以外的其他分组。 现在是成为几何群体理论家的激动人心的时刻。 许多美丽的数学专门用于分类结果。 例如，表面的分类（例如球体，与平面相同的球体）是经典的。 此分类类似于Mendeleev的元素周期表 - 每个表面一次出现一次。 近年来，在很大程度上是由于W. Thurston的几何思想，在三个manifolds的分类定理方面取得了长足的进步（当地的空间像3维空间一样）。 受这些多种成功的启发，数学家已经从几何观点开始研究群体。 （一个小组是整个数学和科学的基本对象。粗略地说，一组是某些对象的对称性。）数学已经到达可以设想（几何）组分类定理的可能性。 例如，E。Rips和Z. Sela的最新结果描述了如何将一个小组分成更基本的作品。 在当前项目中，主要研究人员将继续他的研究。 更具体地说，他将以除撕裂和塞拉的方式以其他方式展示如何将小组分解为基本作品。 此外，他将继续与M. Bestvina和M. Handel一起工作，对一些在此分类中发挥关键作用的小组进行详细研究。 ***