Finiteness of mapping class groups: Locally large strongly irreducible Heegaard splittings

By Namazi and Johnson’s results, for any distance at least 4 Heegaard splitting, its mapping class group is finite. In contrast, Namazi showed that for a weakly reducible Heegaard splitting, its mapping class group is infinite; Long constructed an irreducible Heegaard splitting where its mapping class group contains a pseudo anosov map. Thus it is interesting to know that for a strongly irreducible but distance at most 3 Heegaard splitting, when its mapping class group is finite. In [19], Qiu and Zou introduced the definition of a locally large distance 2 Heegaard splitting. Extending their definition into a locally large strongly irreducible Heegaard splitting, we proved that its mapping class group is finite. Moreover, for a toroidal 3-manifold which admits a locally large distance 2 Heegaard splitting in [19], we prove that its mapping class group is finite.