Large-Scale Geometry of Hyperbolic Groups and 3-Manifolds

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DMS-0204506Bruce KleinerThe proposal consists of two projects which aremotivated by the geometry of three dimensional manifolds.The first project uses quasisymmetric homeomorphismsas a framework to understand the asymptotic geometryof hyperbolic groups. Several uniformization and rigidity problems are discussed. The central problemis Cannon's conjecture that a hyperbolic group withboundary homeomorphic to the two-sphere is a latticein the isometry group of hyperbolic space. A resolutionof this conjecture would be a major step toward Thurston'sHyperbolization conjecture. The second part of the proposal addresses the Weak hyperbolization conjecture,the other missing step in the proof of Thurston's conjecture.The proposed attack uses geometry and dynamics ofsurface laminations in three manifolds.One of the major open problems in Mathematics is to determine the possible shapes (topologies) that a three dimensional space might assume. In the 1970's, Thurston formulated his famousGeometrization conjecture, which is a precise statement abouthow the classification of such spaces should look. The proposal addresses a piece of Thurston's conjecture known as the Hyperbolization conjecture. The general line of attack is to associate a two dimensional self-similar fractal sphere with each three dimensional space, and reduce the originalconjecture to showing that one can find "good" coordinate systemsfor this associated fractal sphere.