On the group action of Mod(M,F) on the disk complex

Let (V,W;F) be a weakly reducible, unstabilized, Heegaard splitting of genus at least three in an orientable, irreducible 3-manifold M. Then Mod(M,F) naturally acts on the disk complex D(F) as a group action. In this article, we prove if F is topologically minimal and its topological index is two, then the orbit of any element of D(F) for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of D(F) whose orbits are finite if the genus of F is three.