Mathematical Sciences: Studies of Negatively Curved Groups

这个项目试图证明一个负弯曲群的猜想，其无穷远处的视球是2球，在实双曲三维空间上是紧的、适当不连续的、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-斯文森的一个结果将这个猜想简化为证明在无穷远处视球的一个特定的带状序列是共形的。在这种情况下，Cannon-Floyd-Parry的结果将两个一致性公理简化为一个比它们中的任何一个都弱的公理。这就进一步简化了问题，证明了有限环空集合的组合模对这个环空序列具有从0一致有界的组合模。研究这些带状结构的一种很有前途的方法是扩展复合体和扩展图。在双曲的情况下，膨胀复合体对应于星象，而膨胀图对应于缩小到更小的星象。如果支链序列不共形，则基于不同点的膨胀复合体应显示出无穷小的畸变。将扩展复形与无穷远处球面的子集相识别，使人们能够将基于不同点的扩展复形联系起来，因此，无穷小的扭曲可以提供“线场”的替代品。计算的3流形是检验这一点的一个很好的起点。低维几何和拓扑学的一个中心问题是几何在多大程度上支配了三维拓扑。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）可以自然地分解成可以配备几何结构的块。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方法。该项目是在负弯曲空间的主要情况下解决这一猜想的多管齐下的方法的一部分。由于首席研究员和合作者之前的工作，这里的主要研究对象是平面和2球面上的平铺图案。给定具有有限多个模型瓦片的瓦片模式和细分模型瓦片的有限规则，可以递归地细分瓦片模式。问题在于确定瓦片何时存在几何模型，以便在细分时，瓦片的形状保持“接近圆形”（即使它们可能具有分形边界）。这些平铺问题可以用离散共形几何来研究，特别是可以用圆填充来实验研究。圆填充思想和细分思想之间的相互作用非常富有成效，并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. ***