Geometric Group Theory and Surface Dynamics

几何群论和表面动力学

基本信息

批准号：
0706719
负责人：
Michael Handel
金额：
$ 15.41万
依托单位：
Research Foundation Of The City University Of New York (Lehman)
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2007
资助国家：
美国
起止时间：
2007-07-01 至 2011-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0706719&HistoricalAwards=false
关键词：
Geometric Group Theory Surface Dynamics

项目摘要

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.

该提案分为二维动力系统和几何群论相关领域的三个项目。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：对于表面微分同胚群的哪些元素，扶正器具有带有全局不动点的有限索引子群。这直接关系到理解映射类组的哪些子组可以通过微分同胚忠实地作用于表面。我们还在圆盘的面积保持同胚组中寻找卡拉比不变量的类似物，它是为圆盘的面积保持微分同胚定义的。第二个项目是与 Mark Feighn 合作的，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自由群的外自同构群的无限子群要么是可约的，要么包含完全不可约的元素。另一个是证明自由群的自由分裂复形是双曲的。面的映射类群、面的微分同胚群和自由群的外自同构群在基本方面是相关的。映射类群元素的分类定理产生有关微分同胚群的子群的代数性质的信息，从而产生对可以作用于曲面的群的种类以及群可以作用的方式的限制。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。该提案的一部分重点是寻找作用在表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来说都是静止的点。另一个是了解一个表面的映射类组在该表面上的动作。该提案的其他部分试图将有关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亚群二分法。前者要求一种算法来确定两个元素是否仅通过坐标变化而有所不同，后者类似于单个元素的基本分类定理的子组。