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表面的MappingClass组的几何形状。 指导校长是这种动作与克莱尼人在双曲空间上的纠正之间的类比。 PI研究的主要主题是凸共聚性（特征和示例），Veech组的广义组合定理（几何结构和理想的边界行为），以及翻译长度的forpseudo-anosov映射类，作用于Teichmuller空间（尤其是在Teichmuller Place上）（尤其是对最小值的质量扩张问题）。 这项工作的各个部分与R. Kent，B。Farb和D. Margalit进行了正在进行的联合项目。表面的塑形（可以将其视为非常弱的表面的自我对称），而映射类组（这是所有这些自我对称的集合）被广泛地研究到脑中。 尽管这些对象主要是对低维拓扑和几何群体理论的兴趣 - 两个领域在过去20至30年中具有显着的增长，但它们也出现在其他领域的贪婪中，包括复杂分析，代数几何学，代数几何，和动力学。 例如，数学中最有趣的空间之一是对配备各种几何或代数结构（所谓的``riemannsurfaces''Moduli空间'的表面进行分类的空间。 该空间在其上的Teichmuller空间和映射类组的触发中编码。 该项目的目的是通过对teichmullerspace的几何形状来更好地理解他们对班级组的某些类别。