A note on the generating sets for the mapping class groups

Abstract

In this short note, we obtain generating sets with two elements for the mapping class group of closed, oriented surfaces of genus three and four, containing elements of the lowest order known so far.

Keywords

• mapping class groups
• generators
• small order
• Dehn twists

References

1. Dehn, M., Die Gruppe der Abbildungsklassen, Acta Math. 69 (1938), 133-206. https://doi.org/10.1007/BF02547712
2. Farb, B., Margalit, D., A primer on mapping class groups, Princeton University Press, (2011).
3. Humphries, S. P., Generators for the mapping class group, In: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), Lecture Notes in Math., 722, Springer, (1979), 44-47. https://doi.org/10.1007/BFb0063188
4. Korkmaz, M., Generating the surface mapping class group by two elements, Trans. Amer. Math. Soc. 357 (2005), 3299-3310. https://doi.org/10.1090/S0002-9947-04-03605-0
5. Korkmaz, M., Minimal generating sets for the mapping class group of a surface, Handbook of Teichmüller Theory 3 (2012), 441-463.
6. Lickorisch, W. B. R., A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769-778. https://doi.org/10.1017/S030500410003824X
7. Wajnryb, B., Mapping class group of a surface is generated by two elements, Topology 35, No.2 (1996), 377-383. https://doi.org/10.1016/0040-9383