| Let M be a closed 3-manifold with a given Heegaard splitting. In this thesis we show that after a single stabilization, some core of the stabilized splitting has arbitrarily high distance with respect to the splitting surface. This generalizes a result of Minsky, Moriah, and Schleimer for knots in S3. We also show that in the complex of curves, handlebody sets are either coarsely distinct or identical. We define the coarse mapping class group of a Heegaard splitting, and show that if (S, V,W) is a Heegaard splitting of genus &ge 2, then CMCG (S, V,W) &cong MCG (S, V,W).This thesis also establishes a new tool for the analysis of link complements, alpha-sloped generalized Heegaard splittings. We examine its relationship to generalized Heegaard splittings of manifolds resulting from Dehn filling. We compare alpha-sloped thin position of 3-manifolds to other types of thin position for knots and 3-manifolds and discuss how this kind of decomposition gives a more organic picture of M and allows the structure of the manifold to dictate the most natural slope on the boundary. Additionally, we provide illustrative examples and questions motivating the study of alpha-sloped thin position. |
