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

文摘：本方案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的改进结果表明，任意Riemann映射的水平集不能任意长。PI和Heinonen正在分析这一结果推广到更一般的从单位圆盘覆盖到多连通区域的映射的确切条件。为此，我们研究了复平面中域的边界的一致厚度，这是一个明显强于一致完备性的条件。第二部分是将Jorgensen不等式推广到作用在当时维单位球面上的离散拟共形群。这样一个推广的Jorgensen不等式将可能把Kleian情形中已知的丰富理论的基本方面推广到拟形式群的设置上。一个自然的问题(例如Gaven Martin)是：在什么假设下离散的拟共形群同构于Klein群？在第三部分中，PI正在研究n维单位球上离散拟共形群的动力学作用问题。这个项目的投资人是爱德华·泰勒。我们研究了离散拟共形群极限集的Hausdorff维的局部性质。例如，我们的问题之一是找出极限集的局部Hausdorff维数和群的局部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，因为它交织在几何、分析和拓扑学中。这项建议是正在进行的一个项目的一部分，该项目旨在研究离散拟共形群的背景下，更广泛地说，在几何函数论中几何与分析的相互作用。对克莱因群(离散的莫比乌斯特拉变换群)的研究可以追溯到18世纪，当时它是由高斯、洛巴切夫斯基、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Kleian群(以及更一般的集合)的混沌行为集合的厚度，并研究在什么假设下这类集合是一致厚的。另一个目标是探索某些解析和几何定义的性质如何随着Kleian群的类别的扩大而变化。我们主要感兴趣的扩大的群类是离散拟共形群类。一个目标是分析如何量化拟共形群类中的离散性概念，另一个目标是将拟共形群在双曲空间边界上的共形作用与其在双曲空间上的作用联系起来。我们的许多工作都是受到类似的猜想和双曲几何领域的发展的启发。

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.