Topology and the Fundamental Group

9803868Cannon是三维空间中最重要的不变量，它的基本群。这位研究人员和他的同事正在通过由这些群定义的无穷远处的递归特征渐近地研究三维流形的基本群。这种方法将关于三维空间的难题转化为关于无限大的二维球面上的递归拼接图案的难题。所得到的图案在美学上非常吸引人，似乎支持了三维流形理论、三维流形几何、几何和组合群理论、圆填充、经典复变量理论、Teichmueller空间理论和Kleinian和Fuchsian群理论之间的深刻联系，以及与迭代有理映射理论、平铺理论、Blaschke积、Grothendieck dessins d‘enfants等理论的潜在联系。最初的理论发展本质上只有一个目的，即证明一个猜想：一个离散群是克莱因的当且仅当它在大的(在格罗莫夫意义下)是负弯曲的，并且有二维球面作为其无穷大的空间。这个问题的肯定解决方案将填补瑟斯顿程序中的一个重要空缺，表明所有的三维流形都具有几何结构。平面和两维空间的递归平铺理论非常丰富和有价值，很可能会有远远超出其预期目标的应用。离散群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，在大空间上是负弯曲的(Gromov和Olshanskii)，并且在无穷远处具有递归的自相似结构(Cannon)。一种可能的说法是，泛型群体将碎片阴影投射到无限远的地方。这位研究人员和他的同事们正在利用共形映射理论来优化这些阴影的几何形状，从而研究这种无限大的分形结构。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，并有可能应用于生物细胞生长理论。

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.***