Mathematical Sciences: Real Trees and Free Groups

闭曲面的映射类群作用于紧化的Teichmuller空间。这个空间的拓扑结构，以及这个动作的动态，可以很好地理解，并提供有关映射类组的全局属性的信息。例如，它满足了Tits的选择。本课题是通过自由群的外自同构群对紧化外空间的作用，继续进行类似的研究。外空间中的点是简单度量树的等价类。紧化点是由简单树的退化得到的实体树。群体在物理科学中大量存在，因为它们是作为一个物体的一组对称性而出现的。有一个特殊的群体，叫做自由群体，所有其他群体都可以从这个群体中构造出来。该项目将继续调查自由群体的基本结构。该技术使用的工具是由E. Rips最近开发的，并由m.b Bestvina的研究者推广。

The mapping class group of a closed surface acts on compactified Teichmuller space. The topology of this space, as well as the dynamics of this action, are well understood and give information about global properties of the mapping class group. For example, it satisfies the Tits alternative. The project is to continue an analogous study of the outer automorphism group of the free group via its action on compactified outer space. Points in Outer Space are equivalence classes of simplicial metric trees. The compactifying points are classes of real trees, objects obtained from degenerations of simplicial trees. Groups abound in the physical sciences because they arise as the set of symmetries of an object. There is a special group, called the free group, from which all others may be constructed. The project is to continue the investigation of the basic structure of the free group. The technique is to use tools recently developed by E. Rips and generalized by the investigator with M. Bestvina.