Mapping Class Groups, Teichmuller Spaces and Low Dimensional Topology

主要研究者：Howard MasurMasur将研究几个涉及映射类群和teichmuller理论的课题。他将完成与Yair Minsky关于映射类群的组合群论的合作工作。这项工作包括求解共轭问题和寻找双自动结构。该方法一直是利用曲线复合体的双曲结构。Masur将开始研究映射类群的大比例尺或渐近几何。本研究是关于Teichmuller空间的拟等距的研究。Masur将继续研究映射类群的表示理论。他将研究一些被称为条件负核的函数，目的是证明该群不具有Kazhdanproperty t。他还将研究表面群是否与映射类群具有同胚性，这与表面上的表面束的研究有关，这是4流形理论中的一个问题。表面拓扑学是研究表面在变形时不改变的性质。该领域与数学中的其他学科（如复分析）以及外部学科（如物理学）相互作用强烈。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研究了这个群体。最近，强大的新几何工具被引入到学习小组中，这是数学中对称的对象。主要研究者正在努力将这些新的几何思想引入曲面和映射类组的研究中。

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.