Mathematical Sciences: Geometric Topology and the Fundamental Group

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9506725 Cannon This project asks, "Does variable negative curvature imply constant negative curvature in dimension three?" More precisely, the project seeks a proof that a negatively curved (Gromov hyperbolic) group with 2-sphere at infinity acts coaompactly, isometrically, and properly discontinuously on hyperbolic 3-space. This goal would constitute a major step forward in establishing Thurston's important geometrization conjecture for 3-manifolds and would also have important consequences in the study of Kleinian groups, negatively curved groups, conformal mapping, and Riemannian geometry. Prior work of the principal investigator and coworkers has reduced the problem to the study of discrete conformal mapping problems about finite subdivision rules in the plane. Secondary, but related, problems concern algorithmic techniques with negatively curved or almost convex groups. All of geometry and its applications takes place within mathematical models or "spaces." Topology seeks to classify these models and to understand their chief local and global properties. The most important of these models are the "manifolds," the locally Euclidean spaces. Mathematicians have to great effect long since classified the 2-dimensional manifolds and have a corresponding valuable, yet conjectural, picture of 3-dimensional manifolds, called "Thurston's geometrization conjecture." The project seeks a proof of one piece of that conjecture, namely that 3-manifolds that in the large behave like negatively curved spaces can in fact be smoothed so as to have constant negative curvature in the small. An affirmative answer would reduce many problems about 3-manifolds to well-developed techniques involving matrix theory, algebra, and analysis, with corresponding economies in mathematical physics and elsewhere that these models are used. ***