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

佩雷尔曼的工作解决了瑟斯顿的几何化猜想，从而证实了“大多数”闭三维流形是双曲的。 现在的期望是，鉴于佩雷尔曼的工作，大量的重点3流形拓扑将了解几何和拓扑的有限体积双曲3流形。 受此启发，PI将研究双曲3-流形，它们的基本群及其基本群的表示。这将涉及研究有限的sheeted覆盖空间，有限商群，profinite完成的离散群体，他们的联系与数论，和扩大家庭的图表。PI还将探索其他离散群，如其他李群和映射类群中的格。三维流形在局部上就像我们生活的空间，理解这些对象一直是过去30年研究的中心主题之一。 这些对象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，数学生物学和计算机科学有关。各种各样的代数对象可以与一个三维流形相关联，其中一个（一个群）捕获了流形和与之相关的其他流形的对称性。例如，PI将探索它们与所谓的“扩展图”家族的联系。这些图在计算机科学中很有名，因为它们在构建高效网络中很重要。此外，另一个项目将灵活几何的现代数学世界与可以追溯到古埃及人的初等数论问题联系起来。通过建议的技术解决这个老问题将是非常有趣的。

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.