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、中期和长期的目标。 这些包括证明子群的出（F）要么几乎阿贝尔或有无限维第二界上同调和开发一个类似的出（F）的Masur-明斯基求和公式，适用于映射类组。 Mark Feighn和PI一直专注于Out（F）的共轭问题，在所讨论的元素具有多项式增长的情况下。 其目的是完成这一重要的特殊情况下，然后解决充分共轭问题合并的技术，从多项式增长的情况下，与那些已经开发的解决指数增长的情况。 约翰·弗兰克斯和PI使用相对映射类群方法来研究曲面的自同构的动力学。 一个目标是扩大他们的工作零熵面积保持的球面的超纯曲面的高属。 这个项目涉及的性质，以及之间的关系，三个重要的群体：外自同构群的自由群，映射类组的表面，和sellomorphism组的表面。 拓扑和几何方法用于研究动力系统，反过来，动力学的思想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。 目前有很大的兴趣在大规模的几何组的外部自同构的自由组和这一建议是很好地处于取得实质性进展在这方面。