Geometric Group Theory and Surface Dynamics

几何群论和表面动力学

• 批准号: 1308710
• 负责人: Michael Handel
• 金额: $ 18.54万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308710&HistoricalAwards=false
• 关键词: Geometric; Group; Theory; Surface; Dynamics

There are three sections to the proposal, each the continuation of a long standing collaboration. Lee Mosher and the PI have short, medium and long term goals regarding the outer automorphism group Out(F) of the finite rank free group F. These include proving that subgroups of Out(F) are either virtually abelian or have infinite dimensional second bounded cohomology and developing an analog for Out(F) of the Masur-Minsky summation formula that applies to mapping class groups. Mark Feighn and the PI have been focusing on the conjugacy problem for Out(F) in the case that the elements in question have polynomial growth. The intent is to complete this important special case and then to the solve the full conjugacy problem by merging the techniques from the polynomially growing case with those already developed to solve the exponentially growing case. John Franks and the PI use relative mapping class group methods to study the dynamics of diffeomorphisms of surfaces. One goal is to to extend their work on zero entropy area preserving diffeomorphisms of the sphere to surfaces of higher genus. This project concerns the properties of, and the relationships between, three important groups: the outer automorphism group of a free group, the mapping class group of a surface, and the diffeomorphism group of a surface. Topological and geometric methods are used to study dynamical systems and in return, ideas from dynamics are used to formulate and prove fundamental topological and geometric properties of outer automorphisms and mapping classes. There is currently a great deal of interest in the large scale geometry of the group of outer automorphisms of the free group and this proposal is well situated to make substantial progress on this front.

该提案分为三个部分，每个部分都是长期合作的延续。 Lee Mosher 和 PI 对有限秩自由群 F 的外自同构群 Out(F) 有短期、中期和长期目标。这些目标包括证明 Out(F) 的子群实际上是阿贝尔群或具有无限维第二有界上同调并开发适用于映射类群的 Masur-Minsky 求和公式的 Out(F) 的模拟。 Mark Feighn 和 PI 一直关注 Out(F) 在相关元素具有多项式增长的情况下的共轭问题。 目的是完成这个重要的特殊情况，然后通过将多项式增长情况的技术与已经开发的用于解决指数增长情况的技术合并来解决完整的共轭问题。 John Franks 和 PI 使用相对映射类组方法来研究曲面微分同胚的动力学。 目标之一是将他们在保持球体微分同胚的零熵区域方面的工作扩展到更高属的表面。 该项目涉及三个重要群的属性和之间的关系：自由群的外自同构群、曲面的映射类群和曲面的微分同胚群。 拓扑和几何方法用于研究动力系统，而动力学的思想则用于制定和证明外自同构和映射类的基本拓扑和几何性质。 目前，人们对自由群的外自同构群的大规模几何产生了很大的兴趣，并且该提案非常有可能在这方面取得实质性进展。