Mathematical Sciences: Studies of Negatively Curved Groups

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9704043 Floyd This project is an attempt to prove the conjecture that a negatively curved group whose visual sphere at infinity is the 2-sphere acts cocompactly, properly discontinuously, and isometrically on real hyperbolic 3-space. The proof of this conjecture would be an important step in proving Thurston's Geometrization Conjecture. A result of Cannon-Swenson reduced the conjecture to proving that a particular sequence of shinglings of the visual sphere at infinity is conformal. In this setting, a result of Cannon-Floyd-Parry reduced the two axioms of conformality to a single one that is weaker than either of them. This led to reducing the problem further to proving that a finite collection of annuli have combinatorial moduli bounded uniformly from 0 with respect to this sequence of shinglings. A promising approach to studying these shinglings is that of expansion complexes and expansion maps. Expansion complexes correspond to horospheres in the hyperbolic case, and the expansion map corresponds to shrinking to a smaller horosphere. If the sequence of shinglings is not conformal, expansion complexes based at different points should show the infinitesimal distortions. The identification of the expansion complex with a subset of the sphere at infinity enables one to relate expansion complexes based at different points, and so the infinitesimal distortions may provide a substitute for a "line field." Cubulated 3-manifolds are a good starting point for checking this. A central question in low-dimensional geometry and topology is the extent to which geometry dominates topology in dimension three. Thurston conjectured that 3-manifolds (topological spaces that locally look like Euclidean 3-space) can be naturally decomposed into pieces that can be equipped with geometric structures. If this Geometrization Conjecture were true, it would greatly aid the study of these spaces, since the rigidity of the geometry enables one to use much more powerful too ls. This project is part of a multi-pronged approach to settling this conjecture in the dominant case of negatively curved spaces. Because of previous work of the principal investigator and coauthors, the main object of study here is tiling patterns on the plane and on the 2-sphere. Given a tiling pattern with finitely many model tiles and a finite rule for subdividing model tiles, one can recursively subdivide the tiling pattern. The problem is to determine when there are geometric models for the tiles so that under subdivision the shapes of the subtiles stay "almost round" (even though they may have fractal boundaries). These tiling problems can be studied using discrete conformal geometry, and, in particular, can be studied experimentally using circle packings. The interplay between the circle packing ideas and the subdivision ideas has been very fruitful and suggests that the methods being developed here could provide useful algorithmic techniques for studying geometrical tiling problems. ***