Geometric Group Theory and Surface Dynamics

本课题在二维动力系统和几何群论的相关领域分为三个课题。第一个是与约翰·弗兰克斯合作，研究表面上光滑的群体作用。一个目标是理解阿贝尔群的一个动作何时必须有一个全局不动点。另一个是证明某些格在二维球面上基本上没有非平凡的作用。第二个项目是与Benson Farb合作。本课题的一部分是研究曲面的映射类群到该曲面的微分同构群的提升。第三个项目继续与Lee Mosher合作。映射类群在Teichmuller空间上的作用与外自同构群在外空间上的作用是平行的。这次合作的目标是在外层空间中制定一个类似的Teichmuller测地线。曲面的映射类群、曲面的微分同构群和自由群的外自同构群之间存在着基本的联系。映射类群元素的分类定理给出了微分同构群的子群的代数性质的信息，从而给出了可以作用于曲面的群的种类和群的作用方式的限制。研究外部自同构群的主要动机之一是它与映射类群的密切联系。该提案的一部分侧重于寻找全球固定的行动点；即对于与群相关的每一个微分同构都是平稳的点。另一个是理解曲面的映射类组在该曲面上的动作。要了解一个群的几何，就必须了解它的测地线；也就是任意两点之间的最短路径。映射类组的测地线已经被很好地理解了一段时间。该项目的第三个目标是从已知的映射类组的测地线中进行推广，以识别和研究外部自同构组的测地线。

The proposal divides into three projects in the related fields of two dimensional dynamical systems and geometric group theory. The first is a collaboration with John Franks to study smooth group actions on surfaces. One goal is to understand when an action by an abelian group must have a global fixed point. Another is to prove that certain lattices have essentially no non-trivial actions on the two dimensional sphere. The second project is joint work with Benson Farb. One part of this project is to study lifts of the mapping class group of a surface to the diffeomorphism group of that surface. The third project continues joint work with Lee Mosher. The action of the mapping class group on Teichmuller space is paralleled by the action of the outer automorphism group on Outer Space. The goal of this collaboration is to formulate an analogue in Outer Space of Teichmuller geodesics. The mapping class group of a surface, the diffeomorphism group of a surface and the outer automorphism group of the free group are related in fundamental ways. Classification theorems for elements of the mapping class group yield information about the algebraic properties of subgroups of the diffeomorphism group and so yield restrictions on the kinds of groups that can act on surfaces and on the ways in which groups can act. One of the main motivations for studying the outer automorphism group is its very close connection to mapping class groups. One part of the proposal focuses on finding global fixed points of actions; i.e. points that are stationary for every diffeomorphism associated to the group. Another is to understand actions of the mapping class group of a surface on that surface. To understand the geometry of a group one must understand its geodesics; i.e. what the shortest paths are between any two points. The geodesics of the mapping class group have been well understood for some time. A third goal of the project is to generalize from what is known about the geodesics of the mapping class group to identify and study geodesics for the outer automorphism group.