Algebra and topology of the Johnson filtration

约翰逊过滤的代数和拓扑

• 批准号: 1122020
• 负责人: Dan Margalit
• 金额: $ 1.15万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2011-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1122020&HistoricalAwards=false
• 关键词: Algebra; topology; Johnson; filtration

In order to more deeply understand the algebraic structure of the mapping class groups, the PI will study the Torelli group, which is the kernel of the symplectic representation. More generally, the PI will investigate the Johnson filtration, which is a sequence of subgroups of the mapping class group starting with the Torelli group. A major open question is whether or not Torelli groups are finitely presented or not. The Pi would also like to understand how pseudo-Anosov dilatations are distributed among the terms of the Johnson filtration. Finally, the PI aims to find generating sets for terms of the Johnson filtration and related groups.The mapping class group describes the symmetries of surfaces, or two-dimensional objects which are analogous to the three-dimensional space we live in. The set of symmetries of an object is often most easily understood by using matrices. In the case of the mapping class group, there is a natural way to use matrices to understand a large piece of the picture, but there is also a large amount of information that escapes detection by these matrices. It is this mysterious part of the mapping class groups that the PI is studying.

为了更深入地理解映射类群体的代数结构，PI将研究Torelli群体，Torelli群体是合成表示的内核。 更普遍地，PI将研究Johnson过滤，Johnson过滤是从Torelli组开始的映射类组的一系列子组。 一个主要的开放问题是Torelli群体是否有限提出。 PI还想了解如何在Johnson过滤术语中分布伪Anosov扩张。最后，PI的目的是找到约翰逊过滤和相关组的术语的生成集。映射类组描述了表面的对称性或二维对象，这些对象类似于我们所居住的三维空间。通常可以通过使用矩阵来最容易理解对象的对称性。 在映射类组的情况下，有一种自然的方法可以使用矩阵来理解大量图片，但是还有大量信息可以通过这些矩阵来逃避检测。 PI正在研究的是映射课程组的神秘部分。