Geometry, topology and group theory of surfaces

项目负责人：Christopher J. leininger本项目旨在研究曲面上各种结构的几何形状，映射类群在这些空间上的作用，以及曲面同胚的拓扑/动力学方面。这包括(1)与R.P. Kent正在进行的一个关于映射类群中的凸紧性的项目，重点是自由群和面群；(2)映射类群和与D. Margalit密切相关的编织群中代数关系的拓扑研究；(3)在与M. Duchin和K. Rafi的合作项目中，利用测地长度函数研究了表面上某些奇异欧几里得几何结构及其退化；(4)与b的持续项目；Farb和D. Margalit提供了“最小复杂度”曲面同胚的拓扑模型。表面——比如球或甜甜圈的表面——已经被研究了数百年，是数学中基础而美丽的对象。表面的研究本质上是有趣的，但它也负责整个数学领域的创造，以及许多其他技术的发展。因此，曲面及其几何理论是复数分析、微分几何、低维拓扑、几何群论和动力学等几个数学领域的交汇点。许多几何物体可以用表面作为基本的构建块来描述。为了研究这些物体，人们自然会遇到表面的“同胚”概念：这是一种“对称”，它只保留了表面最基本的性质。所有同胚的集合有点笨拙，但是最重要的特性可以被提炼成一个更易于管理的结构，称为“映射类组”。本课题主要研究曲面及其同胚和映射类群的相关问题，以及这些研究对相关几何对象的启示。

AbstractAward: DMS-0905748Principal Investigator: Christopher J. LeiningerThis project aims to study the geometry of various structures onsurfaces, actions of the mapping class group on these spaces, andtopological/dynamical aspects of surface homeomorphisms. Thisincludes (1) an ongoing project with R.P. Kent on convexcocompactness in the mapping class group, with a focus on freegroups and surface groups; (2) a topological investigations ofalgebraic relations in the mapping class group and the closelyrelated braid groups with D. Margalit; (3) a study, via geodesiclength functions, of certain singular euclidean geometricstructures on surfaces and their degenerations in a joint projectwith M. Duchin and K. Rafi; (4) a continuing project withB. Farb and D. Margalit to provide a topological model for"minimal complexity" surface homeomorphisms.Surfaces---like the surface of a ball or a doughnut---have beenstudied for hundreds of years, and are fundamental and beautifulobjects in mathematics. The study of surfaces is intrinsicallyinteresting, but is also responsible for the creation of entirefields of mathematics, as well as the development of techniquesin many others. As such, the theory of surfaces and theirgeometries lies at the juncture of several fields of mathematicsincluding complex analysis, differential geometry,low-dimensional topology, geometric group theory and dynamics.Many geometric objects can be described using surfaces as thebasic building blocks. To study these objects one naturallyencounters the notion of a "homeomorphism" of a surface: this isa kind of "symmetry" which preserves only the most basicproperties of the surface. The set of all homeomorphisms issomewhat unwieldy, but the most important features can bedistilled into a more manageable structure called the "mappingclass group." This project proposes the study of several relatedproblems about surfaces, their homeomorphisms and mapping classgroups, and the implications of these studies to relatedgeometric objects.