Mathematical Sciences: Discrete Groups, Quasiconformal Mappings, and Function Theoretic Properties

9305852 Gehring这个项目涉及与经典复分析及其更现代的拟共形映射的导数以及二维和更高维的Mobius群相关的几何问题的数学研究。关于群的工作主要集中在离散的两个椭圆生成元群的几何问题上。通过从生成元的踪迹得到的复数的三元组来研究群是可能的。但并不是所有的三元组都来自两个生成元群，并且将努力确定确定离散群的集合的描述，或者由三个以上阶的椭圆生成元产生的集合的描述。将这一工作应用于三维完全双曲奥氏球的研究。特别是，在已知的下界中，发现了这些奥布洛德的体积的实质性改进。这个项目的主要目标是证明每一个双曲奥氏球面的体积与对应的群包含大于三阶的椭圆元的体积相同。此值约为.03905...拟共形映射的工作继续研究确定维度大于2的区域何时拟共形等价于相同维度的单位球的基本问题。利用勋飞斯型定理得到了充分条件。如果找到了必要的条件，它们可能必须以某些类型的域的k-连通性来表示。最后，获得该奖项支持的学生将考虑Hardy和Littlewood发现的经典函数理论关系如何延续到有界拟圆盘。几何函数理论的研究集中在如何通过施加各种失真限制的光滑函数将区域映射到另一个区域。哪些领域可以相互转换的问题自然而然地出现了，这种观点自动地导致了具有基本结果的群论问题。***

9305852 Gehring This project is concerned with mathematical research on geometric problems associated with classical complex analysis and its more modern derivatives of quasiconformal mapping and Mobius groups in two and higher dimensions. The work on groups focuses on questions related to the geometry of discrete two generator groups with an elliptic generator. It is possible to study the groups through triples of complex numbers derived from the traces of the generators. But not all triples come from two-generator groups and efforts to determine description of the sets which determine discrete groups or the sets arising from elliptic generators of order greater than three will be undertaken. Applications of this work are made to the study of three- dimensional complete hyperbolic orbifolds. In particular substantial improvements in known lower bounds for the volume of these orbifolds has been found. The primary goal of this project is to show that the volume of every hyperbolic orbifold is the same as those for which the corresponding group contains an elliptic element of order greater than three. This value is approximately .03905... Work in quasiconformal mapping continues on the basic issue of determining when a domain in dimension greater than two is quaiconformally equivalent to the unit ball of the same dimension. Sufficient conditions are known through Schoenflies type theorems. If necessary conditions are found, they will probably have to be formulated in certain types of k- connectivity of the domains. Finally, students supported by this award will consider how classical function theory relations discovered by Hardy and Littlewood carry over to bounded quasidisks. Studies in geometric function theory focus on how domains are mapped into one another by smooth functions on which various distortion restrictions are placed. Questions of which domains can be transformed into one another arise naturally and this point of view automati cally leads to group theoretic questions of fundamental consequence. ***