Large-Scale Geometry of Hyperbolic Groups and 3-Manifolds

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

DMS-0204506 Bruce Kleiner The proposal consists of two projects which are motivated by the geometry of three dimensional manifolds. The first project uses quasisymmetric homeomorphisms as a framework to understand the asymptotic geometry of hyperbolic groups. Several uniformization and rigidity problems are discussed. The central problem is Cannon's conjecture that a hyperbolic group with boundary homeomorphic to the two-sphere is a lattice in the isometry group of hyperbolic space. A resolution of this conjecture would be a major step toward Thurston's Hyperbolization 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 of surface 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 famous Geometrization conjecture, which is a precise statement about how 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 original conjecture to showing that one can find "good" coordinate systems for this associated fractal sphere.