Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings
Yanqing Zou
East China Normal University, Shanghai, ChinaRuifeng Qiu
East China Normal University, Shanghai, China
![Finiteness of mapping class groups: locally large strongly irreducible Heegaard splittings cover](/_next/image?url=https%3A%2F%2Fcontent.preview.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-ggd-volume-14-issue-2.png&w=3840&q=90)
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