Geometry, topology and group theory of surfaces

AbstractAward：DMS-0905748首席研究员：Christopher J. Leininger该项目旨在研究表面上各种结构的几何形状，映射类群在这些空间上的作用，以及表面同胚的拓扑/动力学方面。 这包括（1）与R. P.肯特（R. P. Kent）正在进行的关于映射类群中的凸余紧性的研究，重点是自由群和曲面群;（2）与D. Margalit;（3）在与M. Duchin和K. Rafi;（4）与B. Farb和D. Margalit为“最小复杂性”的曲面同胚提供了一个拓扑模型。曲面--就像球或甜甜圈的表面--已经被研究了几百年，是数学中的基础和美丽的分支。 曲面的研究本质上是有趣的，但也是整个数学领域的创造，以及许多其他技术的发展。 因此，曲面及其几何的理论是复分析、微分几何、低维拓扑、几何群论和动力学等几何学领域的交叉点，许多几何对象都可以用曲面作为基本构件来描述。为了研究这些对象，人们自然会遇到曲面的“同胚”概念：这伊萨一种只保留曲面最基本性质的“对称”。 所有同胚的集合有点笨拙，但最重要的特征可以被提炼成一个更易于管理的结构，称为“映射类组”。“本项目提出了几个相关问题的研究曲面，他们的同胚和映射类组，以及这些研究的含义，相关的几何对象。

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.