FRG: Collaborative Research: Understanding Low Volume Hyperbolic 3-Manifolds

这个专题研究小组的目标是证明以下复杂性猜想：完全的低容量双曲三维流形可以通过填充小拓扑复杂性的尖点双曲三维流形来获得。特别是，我们的目标是找到低容量封闭和尖点流形，并解释SnapPea人口普查在确定低容量流形方面的成功。直到20世纪90年代中期，闭可定向双曲3-流形的最佳体积下限似乎是可能的最低体积的1/1000。然后论文“同伦双曲3-流形是双曲的”改善了低容量的界限的一个因素的一百。随后，许多作者利用这一结果来进一步改进下界估计。现在，PI相信他们已经开发出一种基本的新工具（PART技术），它不仅可以找到低容量的闭合和尖点双曲三维流形，而且还可以详细解释为什么复杂性猜想是正确的。我们的方法是一个令人满意的混合初等双曲几何，3流形拓扑，莫尔斯理论，严格的计算机分析。我们的方法的实施将涉及数学家在不同的数学核心领域的专业知识，并与我们的方法中使用的其他领域的良好知识。180年前，W。博尔艾角F. Gauss和N.罗巴切夫斯基开创了一场科学思想的革命，创造了一种替代欧几里得几何的几何学。 这种非欧几里德几何，称为双曲几何，已被证明是数学中的一个了不起的工具。 例如，W. Thurston在1970年代和1980年代表明，绝大多数三维空间（3-流形）具有以双曲几何为模型的几何结构，并且这种几何结构可以用来回答关于基本三维流形的基本问题。 事实上，双曲三维流形在过去的40年里一直是严格审查的主题，取得了惊人的成果;最近，Y。尽管有这些进步和G.佩雷尔曼，该理论的最基本要素之一仍有待理解。特别地，分析双曲三维流形的最自然的工具是使用几何来测量其大小，即，Thurston证明了存在一个最小体积的双曲3-流形，一个次低体积，一个次低体积，等等，但是尽管经过25年的努力，没有一个具有这些低体积的3-流形被最终确定。 这个提议引入了一个令人吃惊的新技术--双曲技术--PI计划开发这个技术来找到所有这些低容量流形，并解释低容量双曲流形必须具有的属性。

The goal of this Focused Research Group is to prove the following Complexity Conjecture: that the complete low-volume hyperbolic 3-manifolds can be obtained by filling cusped hyperbolic 3-manifolds of small topological complexity. In particular, our goal is to find the low-volume closed and cusped manifolds and to explain the success of the SnapPea census in determining the low-volume manifolds. Up to the mid 1990's the best lower bounds for volume of closed orientable hyperbolic 3-manifolds appeared to be approximately 1/1000 of the likely lowest volume. Then the paper "Homotopy Hyperbolic 3-Manifolds Are Hyperbolic" improved the low-volume bounds by a factor of one hundred. Subsequently, many authors have used this result to achieve further improvements in the lower bound estimate. Now, the PI's believe they have developed a fundamental new tool (the MOM technology) which will not only find the low-volume closed and cusped hyperbolic 3-manifolds, but also explain in sharp detail why the Complexity Conjecture is correct. Our method is a satisfying mix of elementary hyperbolic geometry, 3-manifold topology, Morse Theory, and rigorous computer analysis. The implementation of our approach will involve mathematicians with expertise in different core areas of math, and with a sound knowledge of the other areas utilized in our methodology. 180 years ago, W. Bolyai, C. F. Gauss, and N. Lobachevsky started a revolution in scientific thought by creating an alternative geometry to Euclidean geometry. This non Euclidean geometry, called hyperbolic geometry, has proven to be a remarkable tool in mathematics. For example, the work of W. Thurston in the 1970's and 1980's showed that the vast majority of 3-dimensional spaces (3-manifolds) possessed geometric structures modeled on hyperbolic geometry, and that this geometric structure could be used to answer fundamental questions about the underlying 3-dimensional manifold. In fact, hyperbolic 3-manifolds have been the subject of intense scrutiny these last 40 years with striking results achieved; most recently, the proofs of the Ending Lamination and Tameness Conjectures, by Y. Minsky et al. Despite these advances and the possible spectacular resolution of the Geometrization Conjecture by G. Perelman, one of the most basic elements of the theory remains to be understood. In particular, the most natural tool for analyzing a hyperbolic 3-manifold is to use the geometry to measure its size, i.e., to compute its volume, but the behavior of the volume function remains mysterious: Thurston proved that there is a least volume hyperbolic 3-manifold, and a next lowest volume, and a next lowest, and so on, but despite 25 years of effort, none of the 3-manifolds possessing these low volumes have been conclusively identified. This proposal introduces a startling new technique--the MOM Technology--that the PIs plan to develop to find all these low-volume manifolds and to explain what properties low-volume hyperbolic manifolds must have.