Geometry and topology of hyperbolic 3-manifolds

该项目探讨了三维双曲流形的几何和拓扑结构，这些流形承认黎曼度量具有负截面曲率。关于双曲3-流形的一些最基本的问题涉及到它们的性质在有限次覆盖空间中继承的程度，或者相反，在有限次覆盖空间中可能发展的程度。例如，著名的虚拟哈肯猜想和虚拟光纤猜想回答了这一描述，下面的问题也是如此。给定一个具有特殊拓扑性质的双曲3流形，例如一个结补，它与多少其他具有相同性质的双曲3流形共享一个有限次覆盖？或者对于一个固定的双曲3流形M和一组有限次覆盖{M_n}， M_n的基本群的秩是否随覆盖度线性增长？这涉及到拓扑动力学研究中的固定价格问题。PI打算继续对这些问题和其他问题进行攻击，例如嵌入右翼马丁团体的可能性。他还将进一步描述小体积双曲流形的拓扑结构。这里研究的对象，3流形，是在每个点的附近，看起来像我们生活的熟悉的三维空间的空间。尽管它们的定义允许相当大的灵活性，但3-流形与2-流形（曲面）共享一个特性，即每个流形都有一个唯一的最佳度量，一种测量空间中两点之间距离的方法。低维流形的这种性质，在3维中被称为瑟斯顿的几何化猜想，最近被V.佩雷尔曼证明了，它激发了大量的工作，这些工作使用几何作为理解3维流形的工具，该项目遵循了这个广泛的主题。3流形的研究得益于它与许多不同数学领域的相互作用，该项目还借鉴了拓扑动力学、几何流、几何群论等研究中的问题和技术。这个项目将帮助我们更好地理解3流形的分类以及它们之间的关系。这反过来又应用于许多不同的领域，如宇宙学（例如确定宇宙的形状）、生物学（理解DNA的结）和计算机科学（通过与某些3流形族大致相似的图族的联系）等等。

The project explores the geometry and topology of three-dimensional hyperbolic manifolds, those which admit a Riemannian metric with negative sectional curvatures. Some of the most fundamental questions about hyperbolic 3-manifolds concern the extent to which their properties are inherited in, or, conversely, may develop among covering spaces of finite degree. The famous virtual Haken and virtual fiberering conjectures answer this description, for example, as do the questions below. Given a hyperbolic 3-manifold with a special topological property, for instance a knot complement, with how many others that have this same property does it share a finite-degree cover? Or for a fixed hyperbolic 3-manifold M and a family of finite-degree covers {M_n}, does the rank of the fundamental group of M_n grow linearly with the covering degree? This relates to the fixed price question in the study of topological dynamics. The PI intends to continue his attacks on these and other questions, such as the possibility of embedding in right-angled Artin groups. He will also further describe the topology of hyperbolic manifolds with low volume.The objects of study here, 3-manifolds, are spaces which look, in a neighborhood of each point, like the familiar three-dimensional space in which we live. Although their definition allows quite a bit of flexibility, 3-manifolds share with 2-manifolds (surfaces) the property that each admits a unique best metric, a way of measuring the distance between two points in the space. This property of low-dimensional manifolds, known in dimension 3 as Thurston's geometrization conjecture and proven recently by V. Perelman, has motivated an enormous body of work that uses geometry as a tool for understanding 3-manifolds, and the project follows this broad theme. The study of 3-manifolds has benefited from its interaction with many different fields of mathematics, and the project additionally draws from questions and techniques in the study of topological dynamics, geometric flows, geometric group theory, and others. The project will help us better understand the classification of 3-manifolds and the relationships between them. This in turn has applications in such diverse fields as cosmology (in determining the shape of the universe, for example), biology (in understanding the knotting of DNA), and computer science (through connections with families of graphs that certain families of 3-manifolds coarsely resemble), among others.