Mathematical Sciences: Studies in Geometric Group Theory

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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. ***