Mathematical Sciences: Discrete Groups and Quasiconformal Mappings

摘要提案：DMS-9622808 PI：Gehring 这个计画解决有关离散群的莫比乌斯变换、3-流形和轨道、拟共形映射和平面域的函数论性质的问题。 例如，Gehring计划继续与G。马丁的一套价值观，不能假设的迹交换子对的元素的一个非初等离散群。 这项研究已经产生了大量的信息最小可能体积的3-流形和orbifolds和它应该，特别是允许一个证明，orbifold与3-5-3双曲四面体群具有最小体积之间的所有3-流形和orbifolds。本文的主要工具是：（1）用复迭代法来确定一个特殊多项式族P的大填充Julia集E，（2）用多项式族P和填充Julia集E的覆盖论证来获得元素对的扩张子的迹的排除值的大区域D。 格林还希望进一步研究多项式族P，一个有趣的集合多项式与一个复杂的参数自然产生的研究莫比乌斯集团。 数学可以被看作是由三个主要领域-代数，分析和几何-连同许多其他领域的基本思想的发展，在一个或多个这些领域。 因此 代数是从研究多项式方程发展而来的领域，分析是从微积分中发展而来的领域，几何是从欧几里得和希腊人的思想发展而来的领域。 当属于其中一个领域的思想可以用来解决另一个领域的问题时，对数学家来说是特别有趣和令人满意的。 该项目涉及的一些问题， 位于这三个领域交汇的十字路口。 特别地，一个问题是使用复迭代理论确定某些几何对象的最小可能体积， Mandelbrot集的分析与多项式家庭，产生于代数。工作在这样一个部分的数学可以是相当富有成效的，因为解决一个十字路口的问题通常有影响，在所有有关领域。

ABSTRACT Proposal: DMS-9622808 PI: Gehring This project addresses problems concerning discrete groups of Mobius transformations, 3-manifolds and orbifolds, quasiconformal mappings, and function theoretic properties of plane domains. For example, the Gehring plans to continue a joint study with G. J. Martin of the set of values which cannot be assumed by the trace of the commutator of pairs of elements of a nonelementary discrete group. This study has already yielded a great deal of information about the minimum possible volumes of 3-manifolds and orbifolds and it should, in particular, allow one to prove that the orbifold associated with the 3-5-3 hyperbolic tetrahedral group has minimum volume among all 3-manifolds and orbifolds. The main tools for this study are: (1) complex iteration to identify big filled in Julia sets E for a special family P of polynomials, and (2) a covering argument using the family P and the filled in Julia sets E to obtain large regions D of excluded values for the traces of the commutators of pairs of elements. The Gehring hopes also to study further the polynomial family P, an interesting collection of polynomials with one complex parameter which arise naturally in the study of Mobius groups. Mathematics can be viewed as consisting of three main fields - algebra, analysis, and geometry - together with many other areas which are developments of the basic ideas in one or more of these fields. Thus algebra is the field which has developed from studying polynomial equations, analysis the field which has its roots in the calculus, and geometry the field that has developed from the ideas of Euclid and the Greeks. It is particularly interesting and satisfying for mathematicians when ideas which belong to one of these fields can be used to solve problems in another. This project concerns some problems which are situated at a crossroads where all three of these fields meet. In particular, one problem is to determine the smallest possible volume of certain g eometric objects using the theory of complex iteration and Mandelbrot sets of analysis together with a polynomial family which arose from algebra. Work in such a part of mathematics can be quite fruitful since the solution of a crossroads problem usually has implications in all of the fields concerned.