Mathematical Sciences: Real Trees and Free Groups

闭曲面的映射类群作用于紧化的TeichMuller空间。该空间的拓扑以及该动作的动态被很好地理解，并提供关于映射类组的全局属性的信息。例如，它满足了Tits的选择。该项目将通过自由群在紧致外层空间上的作用，继续对自由群的外自同构群进行类似的研究。外层空间点是单纯度量树的等价类。紧化点是实树的类，是由单纯树的退化得到的对象。群在自然科学中比比皆是，因为它们是作为一组物体的对称性出现的。有一个特殊的群体，称为自由群体，所有其他的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团体的基本结构。该技术是使用E.Rips最近开发的工具，并由M.Bestvina的研究人员推广。

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.