Topology and the Fundamental Group

拓扑和基本群

• 批准号: 9803868
• 负责人: James Cannon
• 金额: $ 7.38万
• 依托单位: Brigham Young University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1998
• 资助国家: 美国
• 起止时间: 1998-08-01 至 2001-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9803868&HistoricalAwards=false
• 关键词: Topology; Fundamental; Group

9803868Cannon The most important invariant of three-dimensionalspaces is the fundamental group. The investigator and hiscoworkers are studying the fundamental group of three-dimensional manifolds asymptotically via the recursivepatterns at infinity defined by those groups. This approachtransfers difficult problems about three-dimensional spacesto difficult problems about recursive tiling patterns in the two-dimensional sphere at infinity. The patterns obtained are very attractive aesthetically and seem tosupply deep connections among three-manifold theory, thegeometry of three-manifolds, geometric and combinatorialgroup theory, circle packing, classical complex variabletheory, Teichmueller space theory, and the theory of Kleinianand Fuchsian groups, with potential connections as well tothe theories of iterated rational maps, tiling theory, Blaschke products, Grothendieck's dessins d'enfants, etc.The initial theory was developed with essentially only oneaim in mind, namely to prove the conjecture that a discretegroup is Kleinian if and only if it is negatively curved in the large (in the sense of Gromov) and has the two-dimensionalsphere as its space at infinity. An affirmative solution to thisproblem would fill an important slot in Thurston's programto show that all three-manifolds admit a geometric structure.The theory of recursive tilings of the plane and two-sphereis very rich and rewarding and is likely to have applicationswell beyond its intended target. Discrete group theory can be used to model physical motion,geometric symmetries, and biological cell growth. The genericdiscrete group is infinite and negatively curved in thelarge (Gromov and Olshanskii) and has a recursive self-similarity structure at infinity (Cannon). One mightalternatively say that the generic group casts fractalshadows at infinity. The investigator and his coworkersare studying this fractal structure at infinity by usingthe theory of conformal mappings to optimize the geometricshape of these shadows. The results have application to the study of discrete groups, three-dimensional spaces,complex variables, computational group theory, and, potentially,to the theory of biological cell growth.***

小行星9803868 三维空间最重要的不变量是基本群。 研究者和他的同事们正在研究三维流形的基本群，通过这些群定义的无穷远递归模式来渐近地研究。 这种方法将三维空间的困难问题转化为无穷远处二维球面上的递归平铺模式的困难问题。 所得到的模式是非常有吸引力的美学和似乎供应深的连接之间的三流形理论，thegathes的三流形，几何和组合群理论，圆包装，经典复变理论，Teichmueller空间理论，和理论的Kleinian和Fuchsian群，与潜在的连接以及理论的迭代有理映射，平铺理论，Blaschke产品，Grothendieck的dessins d'enfants，最初的理论的发展基本上只有一个目的，即证明猜想，一个离散群是Kleinian当且仅当它是负弯曲的大（在意义上的格罗莫夫），并有二维sphere作为其空间在无穷远。 一个肯定的解决这个问题将填补一个重要的插槽Thurston的计划，以表明所有的三流形承认一个几何结构的理论递归tilings的平面和两个sphereis非常丰富和有益的，很可能有applicationswell超出其预期的目标。 离散群理论可以用来模拟物理运动，几何对称性和生物细胞生长。 一般离散群是无限的，在大范围内是负弯曲的（Gromov和Olshanskii），并且在无穷远处具有递归自相似结构（Cannon）。 也可以说，类属群在无穷远处投射分形阴影。 研究者和他的同事们正在研究这种在无穷远处的分形结构，通过使用保角映射理论来优化这些阴影的几何形状。 这些结果可应用于离散群、三维空间、复变量、计算群论的研究，并可能应用于生物细胞生长理论。