Geometric Group Theory and Surface Dynamics

几何群论和表面动力学

• 批准号: 0405814
• 负责人: Michael Handel
• 金额: $ 11.16万
• 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: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-15 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405814&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 a collaboration with John Franks to study smooth group actions on surfaces. One goal is to understand when an action by an abelian group must have a global fixed point. Another is to prove that certain lattices have essentially no non-trivial actions on the two dimensional sphere. The second project is joint work with Benson Farb. One part of this project is to study lifts of the mapping class group of a surface to the diffeomorphism group of that surface. The third project continues joint work with Lee Mosher. The action of the mapping class group on Teichmuller space is paralleled by the action of the outer automorphism group on Outer Space. The goal of this collaboration is to formulate an analogue in Outer Space of Teichmuller geodesics. 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 is its very close connection to mapping class groups. One part of the proposal focuses on finding global fixed points of actions; i.e. points that are stationary for every diffeomorphism associated to the group. Another is to understand actions of the mapping class group of a surface on that surface. To understand the geometry of a group one must understand its geodesics; i.e. what the shortest paths are between any two points. The geodesics of the mapping class group have been well understood for some time. A third goal of the project is to generalize from what is known about the geodesics of the mapping class group to identify and study geodesics for the outer automorphism group.

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·弗兰克斯合作研究表面上的平滑群体动作。 一个目标是了解阿贝尔群的动作何时必须具有全局不动点。 另一个是证明某些晶格在二维球体上本质上没有非平凡的作用。 第二个项目是与 Benson Farb 的合作。 该项目的一部分是研究表面的映射类群到该表面的微分同胚群的提升。 第三个项目继续与 Lee Mosher 合作。 映射类群在 Teichmuller 空间上的作用与外层自同构群在外层空间上的作用是平行的。 此次合作的目标是在外太空中建立泰希米勒测地线的模拟。 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全局固定的行动点；即对于与该群相关的每个微分同胚来说都是静止的点。 另一个是了解一个表面的映射类组在该表面上的动作。 要了解群的几何形状，必须了解其测地线；即任意两点之间的最短路径是什么。 一段时间以来，绘图类组的测地线已被很好地理解。 该项目的第三个目标是根据映射类组的测地线的已知信息进行概括，以识别和研究外部自同构组的测地线。