Let M be a closed 3-manifold with a given Heegaard splitting. 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 S^3. 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 Heeegaard
splitting, and show that if (S, V, W) is a Heegaard splitting of genus greater
than or equal to 2, then the coarse mapping class group of (S,V,W) is
isomorphic to the mapping class group of (S, V,W).