Mathematical Sciences: Studies in Geometric Group Theory

数学科学：几何群论研究

• 批准号: 9400900
• 负责人: William Floyd
• 金额: $ 6.36万
• 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: 1994
• 资助国家: 美国
• 起止时间: 1994-07-15 至 1997-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9400900&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Studies; Geometric; Group

9400900 Floyd Professor Floyd plans to continue a joint research project, with Professor J. W. Cannon (of Brigham Young University) and Professor W.R. Parry (of Eastern Michigan University), in geometric group theory. Their research is an attempt to prove the conjecture that a word hyperbolic group with visual sphere at infinity the 2-sphere acts cocompactly, properly discontinuously, and isometrically on real hyperbolic 3-space. By a result of Cannon-Swenson, proving the conjecture is equivalent to proving that a particular shingling of the visual sphere at infinity is conformal. The main emphasis of the project is on determining when a sequence of shinglings of a surface is conformal. An important special case is to understand the intrinsicgeometry of a sequence of tilings given by a finite subdivision rule. As an example of a finite subdivision rule, consider subdividing a rectangle (tile) by dividing it in half horizontally and into thirds vertically, so as to get six smaller rectangles (subtiles) of equal size. If one repeats the process inductively on the subtiles, one gets new subtiles which are becoming distorted (tall and narrow). Understanding whether one can change the shapes so that the subtiles do not become arbitrarily distorted is at the heart of an important problem in topology and group theory. The investigators are trying to prove that, in the cases under consideration, there is an intrinsic geometry in which the subtiles do not become distorted. In some examples, to achieve this one changes the shapes of the tiles so that they have fractal boundaries. ***

小行星9400900弗洛伊德 弗洛伊德教授计划继续一个联合研究项目，与J. W.坎农（杨百翰大学）和W.R.帕里（东密歇根大学），在几何群论。他们的研究是试图证明一个猜想，即一个在无穷远处具有视球面的双曲群2-球面在真实的双曲3-空间上是余紧的、适当不连续的和等距的。 根据Cannon-Swenson的一个结果，证明这个猜想等价于证明在无穷远处视球的一个特定的叠瓦是共形的。 该项目的主要重点是确定当一个表面的叠瓦序列是共形的。 一个重要的特殊情况是理解由有限细分规则给出的平铺序列的内在几何。 作为有限细分规则的一个例子，考虑通过将矩形（瓦片）水平地分成两半并垂直地分成三分之一来细分矩形（瓦片），以便得到六个相等大小的较小矩形（子瓦片）。 如果在子图块上归纳地重复该过程，则得到变得扭曲（高且窄）的新子图块。 了解是否可以改变形状，使subtiles不成为任意扭曲是在拓扑学和群论的一个重要问题的核心。 研究人员试图证明，在所考虑的情况下，有一个内在的几何形状，其中的微妙不成为扭曲。 在一些示例中，为了实现这一点，改变图块的形状，使得它们具有分形边界。 ***