The Interaction Between Geometry and Analysis in Geometric Function Theory and in the Theory of Discrete Groups

摘要：本建议书所述的具体工作包括三个项目。第一部分是与Juha Heinonen的合作项目：Hayman和Wu的一个结果表明，任何Riemann映射的水平集都不能任意长。PI和Heinonen分析了将这一结果推广到多连通域上单位圆盘覆盖映射的更一般情况的精确条件。为此，我们探讨了均匀厚度的域的边界在复杂的planet.Part II是一个项目，包括推广Jorgensen不等式离散拟共形群的单位球上的单位球上的一致perfectness.Condition，是可测量的强。这样一个广义的Jorgensen不等式将使我们有可能把Kleinian情形中已知的丰富理论的基本方面扩展到拟共形群的情形。例如，一个自然的问题（Gaven Martin）是：在什么样的假设下，离散拟共形群与克莱因群同构？在第三部分中，PI正在研究关于离散拟共形群作用于n维单位球的动力学行为的问题。这个项目的一部分是与爱德华泰勒联合。本文研究离散拟共形群极限集的Hausdorff维数的局部性质。例如，我们的问题之一是找到极限集的局部Hausdorff维数与群的局部Poincare指数之间的关系。另一个问题涉及离散群的无限指数子群的极限集。莫比乌斯变换的离散群理论特别美丽，因为它交织着几何，分析和拓扑。这一建议是一个正在进行的计划的一部分，以研究theinteraction几何和分析的设置discrete quasiconformal集团，更普遍的是，在几何函数理论。Kleinian群（离散的高斯变换群）的研究可以追溯到世纪，当时它是由高斯、罗巴切夫斯基、克莱因和庞加莱等数学家发展起来的。我们的目标之一是分析Kleinian群（以及更一般的集合）的混沌行为的集合的厚度，并研究在什么样的假设下这些集合是均匀厚度的。另一个目标是探索如何某些分析和几何定义的性质的变化，因为一个扩大类的克莱因集团。我们主要感兴趣的扩大的群类是离散拟共形群类。一个目的是分析如何量化拟共形群类中的离散性概念，另一个目的是将拟共形群在双曲空间边界上的共形作用与它在双曲空间上的作用联系起来。我们的许多工作都受到了双曲几何领域的类比和发展的启发。

ABSTRACT:The specific work described in this proposal consists of threeprojects. Part I is a joint project with Juha Heinonen: Acelebrated result by Hayman and Wu says that the level sets of anyRiemann mapping can not be arbitrarily long. The PI and Heinonenare analyzing the exact conditions under which this result extendsto the more general case of covering maps from the unit disk ontomultiply connected domains. To this end, we explore the uniformthickness of boundaries of domains in the complex plane, acondition that is measurably stronger than uniform perfectness.Part II is a project that consists of generalizing Jorgensen'sinequality to discrete quasiconformal groups acting on then-dimensional unit sphere. Such a generalized Jorgensen inequalitywould make it possible to extend fundamental aspects of the richtheory known in the Kleinian case to the setting of quasiconformalgroups. A natural question (with Gaven Martin) for example is:Under what assumptions is a discrete quasiconformal groupisomorphic to a Kleinian group? In part III, the PI is working onquestions concerning the dynamical action of a discretequasiconformal group acting on the n-dimensional unit ball. Aportion of this project is joint with Edward Taylor. We areexploring local properties of the Hausdorff dimension of limitsets of discrete quasiconformal groups. One of our questions is,for example, to find the relation between the local Hausdorffdimension of the limit set and the local Poincare exponent of thegroup. Another question involves limit sets of infinite indexsubgroups of discrete groups.The theory of discrete groups of Mobius transformations isespecially beautiful as it intertwines geometry, analysis, andtopology. This proposal is part of an ongoing program to study theinteraction of geometry and analysis in the setting of discretequasiconformal groups and more generally, in geometric functiontheory. The study of Kleinian groups (discrete groups of Mobiustransformations) goes back to the 18th century, when it wasdeveloped by such mathematicians as Gauss, Lobachevsky, Klein, andPoincare. One of our goals is to analyze the thickness of the setof chaotic behavior of a Kleinian group (and more general sets)and to investigate under what assumptions such sets are uniformlythick. Another goal is to explore how certain analytically andgeometrically defined properties change as one enlarges the classof Kleinian groups. The enlarged class of groups that we aremainly interested in is the class of discrete quasiconformalgroups. One objective is to analyze how one can quantify theconcept of discreteness in the class of quasiconformal groups.Another goal is to relate the conformal action of a quasiconformalgroup on the boundary of hyperbolic space to its action onhyperbolic space. Much of our work is inspired by analogousconjectures and developments in the field of hyperbolic geometry.