Mathematical Sciences: Studies in Geometric Group Theory

9400900 Floyd Floyd教授计划继续与杨百翰大学的J. W. Cannon教授和东密歇根大学的W. r . Parry教授在几何群论方面的联合研究项目。他们的研究是试图证明在无穷远处具有2球的词双曲群在实双曲三维空间上是紧的、适当不连续的、等距的。根据Cannon-Swenson的一个结果，证明这个猜想等价于证明视球在无穷远处的一个特定的带状是共形的。该项目的主要重点是确定表面的一系列瓦板何时是保形的。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固有几何。作为有限细分规则的一个例子，考虑将一个矩形（tile）进行细分，将其水平分成两半，垂直分成三分之一，从而得到六个大小相等的较小矩形（subtile）。如果一个人在小块上重复这个过程，他会得到新的小块，这些小块会变得扭曲（又高又窄）。了解是否可以改变形状，使细微块不被任意扭曲，是拓扑和群论中一个重要问题的核心。调查人员正试图证明，在考虑的情况下，存在一种内在的几何结构，其中的细微部分不会变形。在某些例子中，为了实现这一点，我们改变了瓷砖的形状，使它们具有分形边界。＊＊＊

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. ***