Resumen de Involving symmetries of Riemann surfaces to a study of the mapping class group
Grzegorz Gromadzki, Michal Stukow
A pair of symmetries (σ, τ ) of a Riemann surface X is said to be perfect if their product belongs to the derived subgroup of the group Aut+(X) of orientation preserving automorphisms. We show that given g 6= 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 M0 which entirely lives on a Riemann surface and one from M±0 . As a result we obtain the perfectness of the mapping class group Mg 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.
