Geometry and the mapping class group

几何和映射类组

• 批准号: 0603881
• 负责人: Christopher Leininger
• 金额: $ 9.72万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-05-15 至 2009-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603881&HistoricalAwards=false
• 关键词: Geometry; mapping; class; group

This project is aimed at studying the geometry of subgroups of the mappingclass group of a surface acting on the Teichmuller space and other relatedspaces. The guiding principal is an analogy between this action and theaction of a Kleinian group on hyperbolic space. The primary topicsstudied by the PI are convex cocompactness (characterizations andexamples), generalized combination theorems for Veech groups (geometricstructure and ideal boundary behavior), and translation lengths forpseudo-Anosov mapping classes acting on Teichmuller space (especially itsrelation to minimal dilatation questions). Various parts of this workinvolve ongoing joint projects with R. Kent, B. Farb, and D. Margalit.Homeomorphisms of a surface (which can be thought of as self-symmetries ofthe surface in a very weak sense) and the mapping class group (which isthe collection of all such self-symmetries) are widely studied inmathematics. While these objects are primarily of interest in lowdimensional topology and geometric group theory--two fields which haveseen significant growth over the last 20 to 30 years--they also arise in avariety of other areas including complex analysis, algebraic geometry, anddynamics. For example, one of the most intriguing spaces in mathematicsis the space which classifies surfaces equipped with a variety ofgeometric or algebraic structures (the so-called ``moduli space of Riemannsurfaces''). This space is encoded in the Teichmuller space and theaction of the mapping class group on it. The goal of this project is todevelop a better understanding of certain classes of subgroups of themapping class group through the geometry of their action on Teichmullerspace.

该项目旨在研究作用于 Teichmuller 空间和其他相关空间的曲面映射类群的子群的几何形状。 指导原则是该作用与双曲空间上克莱因群的作用之间的类比。 PI 研究的主要主题是凸协紧性（特征和示例）、Veech 群的广义组合定理（几何结构和理想边界行为）以及作用于 Teichmuller 空间的伪阿诺索夫映射类的平移长度（特别是其与最小膨胀问题的关系）。 这项工作的各个部分涉及与 R. Kent、B. Farb 和 D. Margalit 正在进行的联合项目。表面的同胚（可以被认为是非常弱意义上的表面的自对称）和映射类组（它是所有此类自对称性的集合）在数学中得到了广泛的研究。 虽然这些对象主要对低维拓扑和几何群论感兴趣（这两个领域在过去 20 到 30 年中取得了显着增长），但它们也出现在各种其他领域，包括复分析、代数几何和动力学。 例如，数学中最有趣的空间之一是对具有各种几何或代数结构的曲面进行分类的空间（所谓的“黎曼曲面模空间”）。 这个空间被编码在Teichmuller空间和映射类组在其上的作用中。 该项目的目标是通过它们在 Teichmuller 空间上的作用的几何形状，更好地理解映射类群的某些类的子群。