Mathematical Sciences: Studies of Negatively Curved Groups

数学科学：负曲群的研究

• 批准号: 9704043
• 负责人: William Floyd
• 金额: $ 4.32万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9704043&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Studies; Negatively; Curved

9704043 Floyd This project is an attempt to prove the conjecture that a negatively curved group whose visual sphere at infinity is the 2-sphere acts cocompactly, properly discontinuously, and isometrically on real hyperbolic 3-space. The proof of this conjecture would be an important step in proving Thurston's Geometrization Conjecture. A result of Cannon-Swenson reduced the conjecture to proving that a particular sequence of shinglings of the visual sphere at infinity is conformal. In this setting, a result of Cannon-Floyd-Parry reduced the two axioms of conformality to a single one that is weaker than either of them. This led to reducing the problem further to proving that a finite collection of annuli have combinatorial moduli bounded uniformly from 0 with respect to this sequence of shinglings. A promising approach to studying these shinglings is that of expansion complexes and expansion maps. Expansion complexes correspond to horospheres in the hyperbolic case, and the expansion map corresponds to shrinking to a smaller horosphere. If the sequence of shinglings is not conformal, expansion complexes based at different points should show the infinitesimal distortions. The identification of the expansion complex with a subset of the sphere at infinity enables one to relate expansion complexes based at different points, and so the infinitesimal distortions may provide a substitute for a "line field." Cubulated 3-manifolds are a good starting point for checking this. A central question in low-dimensional geometry and topology is the extent to which geometry dominates topology in dimension three. Thurston conjectured that 3-manifolds (topological spaces that locally look like Euclidean 3-space) can be naturally decomposed into pieces that can be equipped with geometric structures. If this Geometrization Conjecture were true, it would greatly aid the study of these spaces, since the rigidity of the geometry enables one to use much more powerful too ls. This project is part of a multi-pronged approach to settling this conjecture in the dominant case of negatively curved spaces. Because of previous work of the principal investigator and coauthors, the main object of study here is tiling patterns on the plane and on the 2-sphere. Given a tiling pattern with finitely many model tiles and a finite rule for subdividing model tiles, one can recursively subdivide the tiling pattern. The problem is to determine when there are geometric models for the tiles so that under subdivision the shapes of the subtiles stay "almost round" (even though they may have fractal boundaries). These tiling problems can be studied using discrete conformal geometry, and, in particular, can be studied experimentally using circle packings. The interplay between the circle packing ideas and the subdivision ideas has been very fruitful and suggests that the methods being developed here could provide useful algorithmic techniques for studying geometrical tiling problems. ***

小弗洛伊德9704043 这个项目是试图证明一个猜想，即一个负曲群的无穷远视球面是2-球面，它作用在真实的双曲3-空间上是余紧的、适当不连续的和等距的。 这个猜想的证明将是证明瑟斯顿几何化猜想的重要一步。 坎农-斯温森的一个结果减少了猜想证明，一个特定的序列shinglings的视觉领域在无穷大是共形的。 在这种情况下，Cannon-Floyd-Parry的一个结果将两个共形性公理简化为一个比它们中任何一个都弱的公理。 这导致减少问题进一步证明，一个有限的集合annuli有组合模界统一从0关于这个序列的叠瓦。 一个有前途的方法来研究这些瓦是膨胀复合体和膨胀地图。 在双曲情况下，膨胀复合体对应于水平球，而膨胀图对应于收缩到更小的水平球。 如果叠瓦序列不是共形的，基于不同点的展开复形应该显示无穷小的畸变。 在无穷远处的球面子集的膨胀复合体的识别使人们能够基于不同点将膨胀复合体联系起来，因此无限小的失真可以提供“线场”的替代。“Cubulated 3-流形是检查这一点的一个很好的起点。 在低维几何和拓扑学中的一个中心问题是几何在三维中支配拓扑的程度。 Thurston指出，3-流形（局部看起来像欧几里得3-空间的拓扑空间）可以自然地分解为可以配备几何结构的部分。 如果这个几何化猜想是正确的，它将极大地帮助这些空间的研究，因为几何的刚性使人们能够使用更强大的工具。 这个项目是一个多管齐下的方法来解决这个猜想在负弯曲空间的主导情况下的一部分。 由于主要研究者和合著者以前的工作，这里的主要研究对象是平面和2-球面上的平铺模式。 给定具有许多模型瓦片的瓦片化模式和用于细分模型瓦片的有限规则，可以递归地细分瓦片化模式。 问题是要确定何时存在瓦片的几何模型，以便在细分下子瓦片的形状保持“几乎圆形”（即使它们可能具有分形边界）。 这些平铺问题可以研究使用离散保形几何，特别是，可以实验研究使用圆填充。 圆包装的想法和细分的想法之间的相互作用是非常富有成效的，并建议，这里正在开发的方法可以提供有用的算法技术，研究几何拼接问题。 ***