A presentation for the mapping class group of a non-orientable surface from the action on the complex of curves (Q935568)

In order to find a presentation of the mapping class group \(\mathcal M(F^n_g)\) of a non-orientable compact surface \(F^n_g\), the author studies the action of \(\mathcal M(F^n_g)\) on the complex of curves \(F^n_g\). The complex of curves of a surface \(F\) is the simplicial complex whose vertices are isotopy classes of non-trivial non-boundary parallel simple closed curves on \(F\) and where \(k+1\) vertices determine a \(k\)-simplex if the corresponding curves admit pairwise non-isotopic disjoint representatives. Using this action a presentation of the mapping class groups has been obtained in the orientable case in \textit{S. Benvenuti} [Adv. Geom. 1, 291--321 (2001; Zbl 0983.57015)]. The author proves that \(\mathcal M(F^n_g)\) can be presented in terms of the isotropy subgroups of collections of curves on \(F^n_g\) (i.e. simplices of the curve complex), provided that \(F^n_g\) is not sporadic, that is the complex of curves of \(F^n_g\) is simply connected. He also proves that a presentation of the isotropy subgroup of a collection of curves \(A\) can be obtained from a presentation of the mapping class group of the surface (eventually disconnected) obtained cutting \(F^n_g\) along \(A\). Moreover, he explicitly determines presentations for the mapping class group in all the sporadic cases. So, using these results it is possible to obtain recursively a presentation of \(\mathcal M(F^n_g)\) for each non-orientable compact surface \(F^n_g\).