Virtual properties of 3-manifold groups

3流形群的虚拟性质

• 批准号: 1308836
• 负责人: Yi Liu
• 金额: $ 10.99万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308836&HistoricalAwards=false
• 关键词: Virtual; properties; manifold; groups

Under the grant, the PI will study topology of 3-manifolds in terms their fundamental groups. The research will be based on recent progress in the Virtual Fibration Conjecture and the Virtual Haken Conjecture. Ingredients about surface subsubgroups in 3-manifolds will be summarized and generalized, and will be applied to approach existing problems including the virtual positivity of representation volumes and the linearity of 3-manifold groups. The project will also study the virtual properties of surface automorphisms. This is closely related to the surface bundle case of Virtual Homological Torsion Conjecture. The PI proposes to apply methods from geometric group theory, dynamical systems, representation theory, and low dimension topology.Spaces and transformations provide an important perspective on mathematically modelling our world. Natural sciences and certain social sciences utilize them as abstract tools to understand symmetry of various systems. The PI will study geometric transformations on 3-dimensional spaces, which form an essential family for many applications involved in physics, computer sciences, and other disciplines. This family of transformations have been proved to be powerful and mathematically comprehensible, thanks to recent solutions of long existing conjectures. The principal pursuit of this project is to attack several existing problems in low dimensional topology with the new techniques. Besides, the project endeavors to theoretically summarize recent development of the area, making the new ideas more accessible to working scientists from all disciplines.

在该资助下，PI将研究三维流形的拓扑结构，其基本群。本文的研究将基于虚纤维化猜想和虚哈肯猜想的最新进展。总结和推广了3-流形中曲面子群的有关内容，并将其应用于研究现有的问题，包括表示体的虚正性和3-流形群的线性性.该项目还将研究表面自同构的虚拟属性。这与虚同调挠猜想的面丛情形密切相关。PI建议应用几何群论，动力系统，表示论和低维拓扑学的方法。空间和变换为我们的世界提供了数学建模的重要视角。自然科学和某些社会科学利用它们作为抽象的工具来理解各种系统的对称性。PI将研究三维空间上的几何变换，这形成了物理学，计算机科学和其他学科中许多应用的基本家庭。这一系列的转换已被证明是强大的和数学上可理解的，由于最近的解决方案的长期存在的代数。本项目的主要目标是利用新技术解决低维拓扑中存在的几个问题。此外，该项目致力于从理论上总结该领域的最新发展，使新的想法更容易为所有学科的工作科学家所接受。