Mathematical Sciences: Discrete Groups and Quasiconformal Mappings

摘要建议：dms-9622808 PI：Gehring这个项目解决了关于Mobius变换的离散群、3-流形和Orborold、拟共形映射以及平面域的函数论性质的问题。例如，Gehring计划继续与G.J.Martin共同研究不能由非初等离散群的元素对的换位子的踪迹来假定的值集。这一研究已经给出了许多关于三维流形和奥比伦流形的最小可能体积的信息，特别是它应该允许人们证明与3-5-3双曲四面体群相关的奥比伦流形在所有的三维流形和奥比伦流形中是最小体积的。本研究的主要工具是：(1)确定特殊多项式族P的大填充Julia集E的复迭代法，以及(2)使用族P和填充Julia集E的覆盖变元，以获得元素对交换子的迹的大区域D的排除值。Gehring还希望进一步研究多项式族P，这是在Mobius群的研究中自然产生的一个带有一个复参数的有趣的多项式集合。数学可以被视为由三个主要领域--代数、分析和几何--以及许多其他领域组成，这些领域是这些领域中的一个或多个基本思想的发展。因此，代数是从研究多项式方程发展而来的领域，分析起源于微积分的领域，几何是从欧几里得和希腊人的思想发展起来的领域。对于数学家来说，当属于这些领域的想法可以用来解决另一个领域的问题时，尤其有趣和令人满意。这个项目涉及一些问题，这些问题处于这三个领域交汇的十字路口。特别地，一个问题是利用复迭代理论和Mandelbrot分析集以及由代数产生的多项式族来确定某些几何对象的最小可能体积。数学中这一部分的工作可能是相当有成效的，因为一个十字路口问题的解决通常涉及到所有有关领域。

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.