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4 1 $begingroup$ Let $Gamma_{g,b}^m$ denote the mapping class group of a genus $g$ surface with $b$ non-permutable parametrised boundary curves and $m$ permutable punctures. It is clear that in general, presentations for these groups are hard. I only need for the general case a set of generators (I want to show that two morphisms $varphi,psi:Gammato G$ are equal and I want to check this on the generators). Partial answers are the following: If $b,m=0$ (so we have a closed surface), then the group is generated by Dehn twists which can be easily drawn on the surface. If $g,m=0$ , we can declare one boundary curve as “outer” and the group is generated by Dehn twists along the inner boundary curves and the pure braid generators $alpha_{ij}$ . From Farb–Margalit, I know that there are always finitely many