# Generators of the mapping class group for surfaces with punctures and boundaries

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 $$\varphi,\psi:\Gamma\to G$$ are equal and I want to check this on the generators). Partial answers are the following:

1. 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.
2. 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 Dehn twists (or half twists) which generate $$\Gamma$$, but can we say in general where they are?

• Thank you, the Gervais paper is really useful! However, it seems not to cover the case with punctures. Of course, we have a surjection $\Gamma_{g, b+m}\to P\Gamma^m_{g, b}$ onto the pure MCG, so we can also describe generators there, but what about $\Gamma_{g, b}^m$ itself? – FKranhold Mar 25 at 14:02
• Okay, so these are $\binom{m}{2}$ additional generators, right? (maybe not all necessary) – FKranhold Mar 25 at 16:06