Involving symmetries of Riemann surfaces to a study of the mapping class group

• Published in: Publicacions matemàtiques 2004, Vol.48 (1), p.103-106
• Main Authors: Gromadzki, G., Stukow, M.
• Format: Article
• Language: English
• Subjects: Automorphisms; Homeomorphism; Mathematical surfaces; Mathematical theorems; Riemann surfaces
• ISSN: 0214-1493; 2014-4350
• DOI: 10.5565/PUBLMAT_48104_04

A pair of symmetries $(\sigma,\tau)$ of a Riemann surface $X$ is said to be perfect if their product belongs to the derived subgroup of the group $\operatorname{Aut}^{+} (X)$ of orientation preserving automorphisms. We show that given $g \neq 2, 3, 5, 7 $ there exists a Riemann surface $X$ of genus $g$ admitting a perfect pair of symmetries of certain topological type. On the other hand we show that a twist can be written as a product of two symmetries of the same type which leads to a decomposition of a twist as a product of two commutators: one from ${\mathcal M}'$ which entirely lives on a Riemann surface and one from ${{\mathcal M}^{\pm}}'$. As a result we obtain the perfectness of the mapping class group ${\mathcal M}_g$ for such $g$ relying only on results of Birman [1] but not on influential paper of Powell [6] nor on Johnson's rediscovery of Dehn lantern relation [3] and nor on recent results of Korkmaz-Ozbagci [4] who found explicit presentation of a twist as a product of two commutators.