Mathematical Sciences: The Geometry of Groups and Real Trees

小行星9626699 在目前的项目中，主要研究者将继续 他的工作，与M。Beststanbul和M.亨德尔，关于 自由群的外自同构群。 所使用的方法是E.里普斯的理论的真实的树木和火车轨道理论的Besteland和韩德尔。 首席研究员还将研究在何种程度上JSJ定理的撕裂和Z。Sela可以扩展到除循环群之外的分裂。 现在是一个令人兴奋的时间是一个几何群论。 许多漂亮的数学都致力于分类结果。 例如，表面（空间，如球面，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-每个表面出现一次且仅出现一次。 近年来，由于W. Thurston，有很大的进展，对分类定理的3-流形（空间是局部像三维空间）。 受到这些成功的启发，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，群是某个对象的对称性的集合。） 数学已经达到了这样一个点，即可以设想（几何）群的分类定理的可能性。 例如，E.里普斯和Z Sela描述了如何将一个组分解为更基本的部分。 在本项目中，主要研究者将继续沿着这些路线进行研究。 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。 此外，他将继续与M。Beststanbul和M.亨德尔对一些在这一分类中起关键作用的群体进行了详细的研究。 ***

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. ***