  We show that if the Hempel distance of a Heegaard splitting is larger than
three then the mapping class group of the Heegaard splitting is isomorphic to a
subgroup of the mapping class group of the ambient 3-manifold. This implies
that given two handlebody sets in the curve complex for a surface that are
distance at least four apart, the group of automorphisms of the curve complex
that preserve both handlebody sets is finite.
