Asymptotic Properties of 3-Manifolds and Their Fundamental Groups

摘要：作者：James W. cannon，我们试图解决三维流形的Thurston几何化猜想的双曲情形。在许多可能的方法中，我们选择研究流形的基本群的渐近递推性质。我们专注于小组定义的无穷远处的渐近带状模式，并寻求证明这些模式是否满足我们在早期研究中引入的充分必要一致性公理的方法。我们在以下三种情况下研究了这一问题：(1)我们研究了具有gromov -双曲群的闭3流形，重点考察了用我们的新扭面配对方法构造的具体例子；(2)研究平面上的一般细分或局部替换规则，在平面上我们有更多的自由来构造具有特殊性质的例子；(3)利用2球研究了2球的分支覆盖，其中对应的问题已经用Thurston的有理映射的组合表征在一定程度上得到了解决。在第三种情况下，与经典Teich- mueller理论的联系变得明显；我们正在寻找适合我们环境的泰奇穆勒理论的一个版本。在所有这些背景下，我们使用KenStephenson的圈包装程序，Epstein, Holt， andRees的自动组程序，以及Jeff Weeks的程序SnapPea，结合提议者和他的同事的程序来构建，几何优化和探索正在研究的模式。威廉·p·瑟斯顿（William P. Thurston）为我们提供了三维数学空间——三维流形——的一幅强有力的猜想图。瑟斯顿的几何化猜想是低维拓扑学中最重要的尚未解决的问题，即使我们撇开球面几何的情况，这个猜想隐含着著名的“百万美元”庞加莱猜想。瑟斯顿认为，每一个三流形都可以以内在的方式分成几个部分，每个部分都以八种自然几何形状中的一种为模型。在每一块，人们可以应用很好理解的代数和几何技术来推导流形的性质。该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最大展开，即取其泛盖，并研究该泛盖的渐近性质。覆盖物可以组合地研究，几乎就像细胞生物学家可以研究植物或动物一样。覆盖物可以用计算机来研究，就像计算机科学家可以研究元胞自动机一样。可以用离散动力系统、微分方程或保角映射的方法对覆盖层进行解析研究。

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.