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).