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

Wesleyan University, Middletown CT

Investigators

Abstract

ABSTRACT: The specific work described in this proposal consists of three projects. Part I is a joint project with Juha Heinonen: A celebrated result by Hayman and Wu says that the level sets of any Riemann mapping can not be arbitrarily long. The PI and Heinonen are analyzing the exact conditions under which this result extends to the more general case of covering maps from the unit disk onto multiply connected domains. To this end, we explore the uniform thickness of boundaries of domains in the complex plane, a condition that is measurably stronger than uniform perfectness. Part II is a project that consists of generalizing Jorgensen's inequality to discrete quasiconformal groups acting on the n-dimensional unit sphere. Such a generalized Jorgensen inequality would make it possible to extend fundamental aspects of the rich theory known in the Kleinian case to the setting of quasiconformal groups. A natural question (with Gaven Martin) for example is: Under what assumptions is a discrete quasiconformal group isomorphic to a Kleinian group? In part III, the PI is working on questions concerning the dynamical action of a discrete quasiconformal group acting on the n-dimensional unit ball. A portion of this project is joint with Edward Taylor. We are exploring local properties of the Hausdorff dimension of limit sets of discrete quasiconformal groups. One of our questions is, for example, to find the relation between the local Hausdorff dimension of the limit set and the local Poincare exponent of the group. Another question involves limit sets of infinite index subgroups of discrete groups. The theory of discrete groups of Mobius transformations is especially beautiful as it intertwines geometry, analysis, and topology. This proposal is part of an ongoing program to study the interaction of geometry and analysis in the setting of discrete quasiconformal groups and more generally, in geometric function theory. The study of Kleinian groups (discrete groups of Mobius transformations) goes back to the 18th century, when it was developed by such mathematicians as Gauss, Lobachevsky, Klein, and Poincare. One of our goals is to analyze the thickness of the set of chaotic behavior of a Kleinian group (and more general sets) and to investigate under what assumptions such sets are uniformly thick. Another goal is to explore how certain analytically and geometrically defined properties change as one enlarges the class of Kleinian groups. The enlarged class of groups that we are mainly interested in is the class of discrete quasiconformal groups. One objective is to analyze how one can quantify the concept of discreteness in the class of quasiconformal groups. Another goal is to relate the conformal action of a quasiconformal group on the boundary of hyperbolic space to its action on hyperbolic space. Much of our work is inspired by analogous conjectures and developments in the field of hyperbolic geometry.