Mapping Class Groups, Teichmuller Spaces and Low Dimensional Topology

摘要建议：DMS-9803497首席研究员：Howard MasurMasur将致力于涉及映射类群和TeichMuller理论的几个主题。他将完成与Yair Minsky在映射类群的组合群论方面的共同工作。这项工作包括解决共轭问题和寻找双自动机结构。该方法一直采用复杂曲线的双曲线结构。Masur将开始研究映射类群的大规模或渐近几何。本研究是关于TeichMuller空间的拟等距的研究。Masur将继续他在映射类群的表示理论方面的工作。他将研究某些称为条件负核的函数，目的是证明群不具有Kazhdan性质T。他还将研究曲面群是否映射到映射类群，这与研究曲面上的曲面丛有关，这是4-流形理论中的一个问题。曲面拓扑学的领域是研究曲面在变形下不变的那些性质。该领域与数学内部的其他学科，如复分析，以及与外部学科，如物理，有很强的相互作用。映射类群是曲面的自映射群，是基本的研究对象。多年来，许多数学家从不同的角度对这个群体进行了研究。最近，一些强有力的新几何工具被引入来研究群，群是数学中对称的对象。这位首席研究员正致力于将这些新的几何概念引入曲面和映射类群的研究中。

AbstractProposal: DMS-9803497Principal Investigator: Howard MasurMasur will work on several topics involving the mapping class group andTeichmuller theory. He will finish his joint work with Yair Minsky onthe combinatorial group theory of the mapping class group. This workincludes solving the conjugacy problem and finding a biautomaticstructure. The method has been to use the hyperbolic structure of thecomplex of curves. Masur will begin a study of the large scale orasymptotic geometry of the mapping class group. This study is relatedto the study of the quasi-isometries of Teichmuller space. Masur willcontinue his work on the representation theory of the mapping classgroup. He will study certain functions called conditional negativekernels, with the goal of proving that the group does not have Kazhdanproperty T. He will also study whether surface groups admithomeomorphisms to the mapping class group, which is related to thestudy of surface bundles over surfaces, a problem in 4-manifoldtheory.The field of surface topology is the study of those properties ofsurfaces that do not change under deformations. The field interactsstrongly with other disciplines within mathematics, such as complexanalysis, and with outside disciplines such as physics. The mappingclass group, which is the group of self mappings of the surface, is afundamental object of study. Many mathematicians over the years havestudied the group from a variety of viewpoints. Recently, powerfulnew geometric tools have been introduced to study groups, which arethe objects of symmetry in mathematics. The principle investigator isworking to bring these new geometric ideas into the study of surfacesand the mapping class group.