# Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings

### Yanqing Zou

East China Normal University, Shanghai, China### Ruifeng Qiu

East China Normal University, Shanghai, China

## Abstract

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.

## Cite this article

Yanqing Zou, Ruifeng Qiu, Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings. Groups Geom. Dyn. 14 (2020), no. 2, pp. 591–605

DOI 10.4171/GGD/556