Geometric Group Theory and Surface Dynamics

几何群论和表面动力学

• 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

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学系统和几何组理论的相关领域的三个项目。 首先是与约翰·弗兰克斯（John Franks）的联合合作。 我们解决的一个问题是：对于表面的差异群的哪个元素，centralizer具有一个有限的索引子组，具有全局固定点。 这直接与理解映射类组的哪些亚组可以通过差异性忠实地在表面上行动。 我们还在寻找保存卡拉比不变式同态同态的区域中的类似物，该磁盘的同态定义为保留磁盘的差异性的区域。 第二个项目的目标是与马克·菲恩（Mark Feighn）的合作，是为了解决自由小组外部自动形态的共轭问题。 第三个项目继续与李·莫瑟（Lee Mosher）进行联合工作。 我们的目标之一是证明自由组的外部自动形态组的无限亚组可以还原或包含完全不可约的元素。 另一个是表明自由组的自由分裂复合物是双曲线的。 表面的映射类组，表面的差异组和自由组的外部自动形态群以基本方式相关。 映射类元素元素的分类定理产生了有关二型差异组亚组的代数特性的信息，因此对可以在表面和组行动的方式上行动的组的类型产生限制。 研究自由组的外部自动形态群体的主要动机之一是它与映射课程组非常紧密的联系。 该提案的一部分集中在寻找作用在表面上的差异的亚组的全球固定点；即子组的每个元素都固定的点。 另一个是了解该表面上表面的映射类组的动作。 该提案的其他部分旨在概括有关映射类群体的已知重要结果，以归为自由组的外部自动形态组。 其中包括共轭问题和亚组二分法。 前者要求一种算法来决定两个元素是否仅因坐标的变化而有所不同，而后者是单个元素基本分类定理亚组的类似物。