Asymptotic Properties of 3-Manifolds and Their Fundamental Groups

摘要奖：DMS-0104030首席研究员：詹姆斯·W·坎农我们试图解决瑟斯顿关于三维流形的几何猜想的双曲型情形。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归性质。我们集中于该群定义的渐近瓦片模式，并寻求证明这些模式满足或不满足我们在前面的研究中引入的充分必要条件的方法。我们在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，重点考察了用我们新的扭面配对方法构造的具体实例；(2)我们研究了平面上的一般细分或局部替换规则，其中我们在构造具有特殊性质的实例时有更多的自由度；(3)2-球面上的Westudy分支覆盖，其中相应的问题已经在某种程度上通过图尔斯顿的有理映射的组合刻画得到了解决。正是在第三个背景下，与经典Teich-Mueller理论的联系变得明显；我们正在寻找适合我们背景的Teichmueller理论的适当版本。在所有这些背景下，我们使用KenStephenson的圈包装程序，Epstein，Holt，Andrees的自动群程序和Jeff Week的程序SnapPea结合提出者和他的同事的程序来构造、几何优化和探索正在研究的模式。威廉·P·瑟斯顿为我们提供了一幅关于三维数学空间-三维流形的强大的猜想图景。瑟斯顿的几何化猜想是低维拓扑学中最重要的悬而未决的问题，即使人们撇开球面几何的情况不谈，其中该猜想蕴含着著名的“百万美元”Poincare猜想。瑟斯顿提出，每个三维流形都可以以一种内在的方式分成几个部分，每个部分都是以八个自然几何中的一个为模型的。在每一块中，人们可以应用熟知的代数和几何技巧来推导流形的性质。这个项目试图解决瑟斯顿猜想的一般情况，即双曲几何的情况。所采用的技巧是最大限度地解开流形，即取其泛覆盖，并研究该覆盖的渐近性质。盖子可以组合起来研究，就像细胞生物学家可以研究植物或动物一样。就像计算机科学家可以研究细胞自动机一样，可以通过计算来研究这一过程。盖子可以用离散动力系统、微分方程组或保角映射的方法进行解析研究。

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.