Topology and the Fundamental Group

三维空间中最重要的不变量是基本群。研究者和他的同事们正通过这些群在无穷远处定义的递归模式渐近地研究三维流形的基本群。这种方法将三维空间的难题转化为二维球面无穷远处递归平铺图案的难题。所得到的模式在美学上非常有吸引力，似乎提供了三流形理论、三流形几何、几何和组合群论、圆填充、经典复变理论、Teichmueller空间理论、Kleinianand Fuchsian群理论之间的深层联系，以及与迭代有理映射理论、平铺理论、Blaschke积、Grothendieck的desdesenfants、最初的理论基本上只有一个目的，即证明一个离散群是克莱因的，当且仅当它在大范围内（在格罗莫夫的意义上）是负弯曲的，并且在无穷远处有二维球面作为它的空间。这个问题的肯定解将填补瑟斯顿计划中的一个重要空缺，表明所有三流形都承认一个几何结构。平面和双球面的递归平铺理论是非常丰富和有益的，并且很可能有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），在无穷远处具有递归的自相似结构（Cannon）。另一种说法是，一般群在无穷远处投射分形阴影。研究者和他的同事正在研究这种分形结构在无穷远处，通过使用保角映射理论来优化这些阴影的几何形状。研究结果可应用于离散群、三维空间、复变量、计算群理论的研究，并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.***