Asymptotic Properties of 3-Manifolds and Their Fundamental Groups

3-流形及其基本群的渐近性质

• 批准号: 0104030
• 负责人: James Cannon
• 金额: $ 10.5万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2001
• 资助国家: 美国
• 起止时间: 2001-07-15 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0104030&HistoricalAwards=false
• 关键词: Asymptotic; Properties; Manifolds; Their; Fundamental

AbstractAward: DMS-0104030Principal Investigator: James W. CannonWe attempt to resolve the hyperbolic case of Thurston'sGeometrization Conjecture for 3-dimensional manifolds. Among themany possible approaches, we choose to study the asymptoticrecursive properties of the fundamental group of the manifold. Weconcentrate on the asymptotic shingling patterns at infinitydefined by the group and seek methods for proving that suchpatterns do or do not satisfy the necessary and sufficientconformality axiom which we introduced in our earlier studies. Westudy this problem in three contexts: (1) We study closed3-manifolds with Gromov-hyperbolic group, with emphasis on theexamination of concrete examples constructed by our new method oftwisted-face-pairings; (2) We study general subdivision orlocal-replacement rules in the plane, where we have more freedomin constructing examples with special properties; and (3) Westudy branched coverings of the 2-sphere by the 2-sphere, wherethe corresponding problem is to some extent already solved bymeans of Thurston's combinatorial characterization of rationalmaps. It is in the third context that the connection withclassical Teich- mueller theory becomes apparent; we are seekingan appropriate version of Teichmueller theory for our setting. Inall of these contexts, we use the circle-packing programs of KenStephenson, the automatic group programs of Epstein, Holt, andRees, and the program SnapPea of Jeff Weeks in conjunction withprograms of the proposer and his coworkers to construct,geometrically optimize, and explore the patterns being studied.William P. Thurston has supplied us with a powerful conjecturalpicture of the spaces of 3-dimensional mathematics, the3-dimensional manifolds. Thurston's Geometrization Conjecture isthe most important unresolved problem in low dimensionaltopology, even if one sets aside the case of spherical geometrywhere the conjecture implies the famous "million dollar" PoincareConjecture. Thurston suggests that every 3-manifold can bedivided in an intrinsic manner into pieces, each of which ismodelled on one of eight natural geometries. Within each piece,one can apply well-understood algebraic and geometric techniquesto derive properties of the manifold. This project seeks toresolve the generic case of the Thurston Conjecture, namely thecase of hyperbolic geometry. The technique employed is tomaximally unwind the manifold, that is, take its universal cover,and study the asymptotic properties of this cover. The cover canbe studied combinatorially almost as a growing cellular organismas a plant or animal might be studied by a cell biologist. Thecover can be studied computationally as a cellular automatonmight be studied by a computer scientist. The cover can bestudied analytically by methods of discrete dynamical systems ordifferential equations or conformal mapping.

摘要奖：DMS-0104030主要研究者：James W.本文试图解决三维流形上Thurston几何化猜想的双曲情形。 在众多可能的方法中，我们选择研究流形的基本群的渐近递归性质。我们集中讨论了由群定义的无穷远处的渐近叠瓦模式，并寻求证明这种模式满足或不满足我们在以前的研究中引入的必要一致性公理的方法。我们从三个方面来研究这个问题：（1）研究具有Gromov-双曲群的闭3-流形，重点是检验用我们的扭面对方法构造的具体例子：（2）研究平面上的一般剖分或局部替换规则，在构造具有特殊性质的例子时有更多的自由;（3）研究了2-球面对2-球面的分枝覆盖，相应的问题已在一定程度上用Thurston的有理映射的组合刻划得到了解决。 正是在第三种情况下，与经典泰希-穆勒理论的联系变得明显;我们正在为我们的环境寻找一个适当的泰希-穆勒理论版本。在所有这些情况下，我们使用KenStephenson的圆包装程序，Epstein，Holt和Rees的自动组程序，Jeff Weeks的程序SnapPea，以及提议者及其同事的程序来构建，几何优化和探索正在研究的模式。William P. Thurston为我们提供了三维数学空间的强大的几何图像，三维流形Thurston的几何化猜想是低维拓扑学中最重要的未解决的问题，即使不考虑球面几何的情况，该猜想隐含着著名的“百万美元”Poincare猜想。Thurston认为，每个三维流形都可以以一种内在的方式分成几部分，每一部分都是以八种自然几何中的一种为模型的。在每一块，可以应用很好理解的代数和几何techniquesto推导流形的属性。这个项目旨在解决瑟斯顿猜想的一般情况，即双曲几何的情况。所采用的技术是对流形进行极大展开，即取其泛覆盖，并研究该覆盖的渐近性质。覆盖层可以被组合地研究，几乎就像细胞生物学家研究一种生长中的细胞有机体，一种植物或动物。这种覆盖可以通过计算来研究，就像计算机科学家研究细胞自动机一样。覆盖可以通过离散动力系统、微分方程或保角映射的方法进行解析研究。