Collaborative Research: Analytic and Geometric Aspects of Convergence Groups

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DMS-0305704Petra Taylor and Ed TaylorThis collaborative proposal details a program that theresearchers are pursuing to extract geometric, analytic anddynamic information concerning discrete groups of quasiconformalmappings acting on a compact metric space. The researchers'program consists of three ``themes'': The first theme concernscertain geometric properties of Kleinian groups that are inducedfrom the existence of the ergodic Liouville measure on the unittangent bundle of a geometrically finite hyperbolic manifold. Aprimary goal is to pair Liouville theory with quasiconformaldeformation theory to the extent that it is revealing of thegeometry of the deformation space of hyperbolic structuresuniformizing a compact co-infinite hyperbolizable three-manifold.The second theme details an ongoing program to show that variousanalytical, dynamical and topological properties of Kleiniangroups (e.g. the exponent of convergence, Hausdorff dimension,porosity, etc.) illuminate the action of the group when the groupin question is generalized to a discrete convergence group. Inthis setting the researchers are primarily concerned withdiscrete quasiconformal groups. The third theme centers aroundcertain non-equivariant interactions between complex analysis andhyperbolic geometry; it generalizes the first two themes. Themotivating question is: How does the asymptotic geometry of theconvex hull boundary of a Jordan curve reflect the conformalgeometry of the curve? Under various stronger assumptions theresearchers are investigating aspects of this question from botha geometric and analytic perspective.The research described in this proposal resides at theintersection between hyperbolic geometry and conformal analysis.These topics form a vast and fundamental area of study inmathematics which dates back to the 18th century, when it wasdeveloped by such mathematicians as Gauss, Lobachevsky, Klein,and Poincare. Though this proposal does not directly addressphysics, we note that both hyperbolic geometry and conformalanalysis (especially in the guise of Teichmuller theory) havefound recent spectacular application in theoretical physics andcosmology. The proposers are dedicated researchers and educators,and Boston College and Wesleyan University both have strong dualidentities as teaching and research institutions. An innovativecomponent of the proposal is to use its multi disciplinary natureto introduce motivated undergraduates to areas of mathematicsthat are currently not well represented (e.g. geometry andanalysis) in the undergraduate degree programs at the proposers'respective institutions. The researchers propose to initiate andmaintain a ``Joint Boston College - Wesleyan University WorkingGroups for Undergraduates and Beginning Graduate Students,'' inpart to entice talented undergraduates into consideration of acareer in mathematical research.