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

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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.