Revistes Catalanes amb Accés Obert (RACO)

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

G. Gromadzki, M. Stukow


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.

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