Mathematical Sciences: The Geometry of Groups and Real Trees

9626699 Feighn在目前的项目中，首席研究员将继续他的工作，与M. Bestvina和M. Handel合作，研究一个自由群的外部自同构群的结构。所使用的方法混合了E. Rips的真树理论和Bestvina和Handel的火车轨道理论。首席研究员还将研究Rips和Z. Sela的JSJ定理在多大程度上可以推广到循环群以外的分裂。现在是成为几何群论家的激动人心的时刻。许多美丽的数学都致力于分类结果。例如，表面（空间，如球体，局部与平面相同）的分类是经典的。这种分类类似于门捷列夫的元素周期表——每个表面出现一次，而且只有一次。近年来，在很大程度上由于W. Thurston的几何思想，在3-流形（局部类似于3维空间的空间）的分类定理方面取得了很大进展。受到这些多方面成功的启发，数学家们开始从几何的角度来看待群。(群是贯穿数学和科学的基本对象。粗略地说，群是某个物体的对称性的集合。数学已经达到了可以设想（几何）群的分类定理的可能性的地步。例如，E. Rips和Z. Sela最近的研究结果描述了如何将一个群体分解成更基本的部分。在当前的项目中，首席研究员将沿着这些思路继续他的研究。更具体地说，他将展示如何以不同于Rips和Sela的方式将群体分解为基本部分。同时，他将继续与贝斯特维纳先生和汉德尔先生合作，对将在这一分类中发挥关键作用的一些群体进行详细研究。* * *

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. ***