Mathematical Sciences: Studies of Negatively Curved Groups

9704043 Floyd 该项目试图证明这样的猜想：一个负弯曲群，其无穷远处的视觉球体是 2 球体，在实双曲 3 空间上协同紧致、适当不连续且等距地起作用。 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-斯文森的结果简化了猜想，证明了无穷远处视觉球体的特定带状疱疹序列是共角的。 在这种情况下，坎农-弗洛伊德-帕里的结果将两个共形公理简化为一个比它们中任何一个都弱的公理。 这导致进一步减少问题，以证明有限的环环集合具有相对于该叠瓦序列从 0 开始均匀限制的组合模量。 研究这些木瓦的一种有前途的方法是扩展复合体和扩展图。 膨胀复合体对应于双曲情况下的星球层，而膨胀图对应于收缩到更小的星球层。 如果木瓦序列不共形，则基于不同点的扩展复合体应显示出无限小的扭曲。 用无穷远球体的子集来识别展开复合体使得人们能够将基于不同点的展开复合体关联起来，因此无穷小的扭曲可以提供“线场”的替代品。 立方体 3 歧管是检查这一点的良好起点。 低维几何和拓扑的一个中心问题是几何在三维中主导拓扑的程度。 瑟斯顿推测，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片段。 如果这一几何化猜想是正确的，它将极大地有助于这些空间的研究，因为几何学的刚性使人们能够使用更强大的工具。 该项目是在负弯曲空间的主导情况下解决这一猜想的多管齐下的方法的一部分。 由于主要研究者和合著者之前的工作，这里的主要研究对象是平面和 2-球体上的平铺图案。 给定具有有限多个模型图块的图块图案和用于细分模型图块的有限规则，人们可以递归地细分图块图案。 问题是确定何时存在图块的几何模型，以便在细分下子图块的形状保持“几乎圆形”（即使它们可能具有分形边界）。 这些平铺问题可以使用离散共形几何来研究，特别是可以使用圆形填料进行实验研究。 圆形堆积思想和细分思想之间的相互作用非常富有成效，并且表明这里开发的方法可以为研究几何平铺问题提供有用的算法技术。 ***

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. ***