Covering spaces of 3-manifolds and representations of their fundamental groups

3-流形的覆盖空间及其基本群的表示

• 批准号: 1105002
• 负责人: Alan Reid
• 金额: $ 29.63万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105002&HistoricalAwards=false
• 关键词: Covering; spaces; manifolds; representations; their

The work of Perelman has resolved the geometrization conjecture of Thurston, thereby confirming that "most" closed 3-manifolds are hyperbolic. The expectation now is that, given Perelman's work, a great deal of the focus of 3-manifold topology will be on understanding the geometry and topology of finite volume hyperbolic 3-manifolds. Motivated by this the PI will study hyperbolic 3-manifolds, their fundamental groups and representations of their fundamental groups. This will involve the study of finite sheeted covering spaces, finite quotient groups, profinite completions of discrete groups, their connections with number theory, and expander families of graphs. The PI will also explore other discrete groups, like lattices in other Lie groups and Mapping Class Groups.Three dimensional manifolds are locally like the space we live in and understanding these objects have been one of the central themes of research in the last 30 years. The importance of these objects extends far beyond their intrinsic interest, since their study connects to mathematical physics, mathematical biology and computer science. Various algebraic objects can be associated to a three dimensional manifold, one of which (a group) captures symmetries of the manifold and other manifolds related to it. Much of the proposal is aimed at exploring properties of these groups. For example, the PI will explore their connections to families of so-called "expanding graphs". These graphs are well-known in computer science because of their importance in building efficient networks. In addition another project connects the modern mathematical world of flexible geometry to a question in elementary number theory that goes back to the ancient Egyptians. A solution to this old question via the techniques suggested would be very interesting.

Perelman的工作解决了Thurston的几何化猜想，从而证实了“大多数”封闭的3个manifolds是双曲线。 现在的期望是，鉴于Perelman的作品，三个manifold拓扑的重点将是理解有限体积的双曲线3个manifolds的几何形状和拓扑。 在此的激励下，PI将研究双曲线3个manifolds，他们的基本群体和基本群体的代表。这将涉及研究有限板的覆盖空间，有限的商组，离散组的完整完成，与数字理论的联系以及图形的范围。 PI还将探索其他离散群体，例如其他谎言组中的格子和映射课程组。三维流形在本地就像我们所居住的空间一样，理解这些对象一直是过去30年来研究的中心主题之一。 这些物体的重要性远远超出了它们的内在兴趣，因为它们的研究与数学物理学，数学生物学和计算机科学联系在一起。各种代数对象可以与三维流形相关联，其中一个（一个组）捕获了歧管的对称性和与之相关的其他歧管的对称性。 大部分建议旨在探索这些群体的特性。例如，PI将探索与所谓“扩展图”家庭的联系。这些图在计算机科学中是众所周知的，因为它们在建立有效的网络中的重要性。此外，另一个项目将柔性几何学的现代数学世界与基本数字理论的问题联系起来，该理论可以追溯到古埃及人。通过建议的技术解决这个旧问题将非常有趣。