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

TL;DR

This paper investigates the action of the mapping class group on the disk complex associated with a Heegaard splitting, revealing conditions under which orbits are infinite or finite, especially for genus three surfaces.

Contribution

It establishes that for topologically minimal surfaces with topological index two, the group action produces infinite orbits, and classifies finite orbits for genus three cases.

Findings

Orbits are infinite for topologically minimal surfaces with index two.

At most two elements have finite orbits when genus is three.

The results depend on the topological minimality and genus of the surface.

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.