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

Abstract: Let $(\mathcal{V},\mathcal{W};F)$ 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 $\mathcal{D}(F)$ 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 $\mathcal{D}(F)$ for this group action consists of infinitely many elements. Moreover, we prove there are at most two elements of $\mathcal{D}(F)$ whose orbits are finite if the genus of $F$ is three.