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·弗兰克斯的合作。我们要解决的一个问题是：对于曲面的微分同构群的哪些元素，扶正器具有具有全局不动点的有限指标子群。这直接关系到理解映射类群的哪些子群可以通过微分同构忠实地作用于一个表面。我们也在寻找圆盘的保面积同胚群中的类似物，Calabi不变量，它是为圆盘的保面积微分同胚定义的。第二个项目是与Mark Feighn合作完成的，其目标是解决自由群的外自同构群的共轭问题。第三个项目继续与Lee Mosher合作。我们的目标之一是证明一个自由群的外自同构群的无限子群是可约的或包含一个完全不可约的元素。另一个是证明自由群的自由分裂复合体是双曲的。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。研究自由群的外自同构群的主要动机之一是它与映射类群的密切联系。该方案的一部分着重于寻找作用于一个曲面上的微分同胚子群的全局不动点；即对于子群的每个元素都是平稳的点。另一个是理解曲面的映射类组在该曲面上的动作。该建议的其他部分试图将关于映射类群的已知重要结果推广到自由群的外部自同构群。这些结果包括共轭问题和一个子群二分法。前者要求一种算法来确定两个元素是否仅通过坐标的变化而不同，后者是对单个元素的基本分类定理的子群的类比。

The proposal divides into three projects in the related fields of two dimensional dynamical systems and geometric group theory. The first is joint work with John Franks. One question that we address is : for which elements of the diffeomorphism group of a surface does the centralizer have a finite index subgroup with a global fixed point. This relates directly to understanding which subgroups of the mapping class group can act faithfully on a surface by diffeomorphisms. We are also looking for an analog in the group of area preserving homeomorphisms of the disk, of the Calabi invariant, which is defined for area preserving diffeomorphisms of the disk. The goal of the second project, which is a collaboration with Mark Feighn, is to solve the conjugacy problem for the group of outer automorphisms of a free group. The third project continues joint work with Lee Mosher. One of our goals is to show that an infinite subgroup of the outer automorphism group of a free group is either reducible or contains a fully irreducible element. Another is to show that the complex of free splittings of the free group is hyperbolic. 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 of the free group is its very close connection to mapping class groups. One part of the proposal focuses on finding global fixed points for subgroups of diffeomorphisms acting on a surface; i.e. points that are stationary for every element of the subgroup. Another is to understand actions of the mapping class group of a surface on that surface. Other parts of the proposal seek to generalize known important results about the mapping class group to the outer automorphism groupof the free group. Among these results are the conjugacy problem and a subgroup dichotomy. The former asks for an algorithm to decide if two elements differ only by a change of coordinates and the latter is the analogue for subgroups of a basic classification theorem for individual elements.