# 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?

## 1 Answer

See the paper

B. Wajnryb, "An elementary approach to the mapping class group of a surface," Geometry & Topology 3 (1999) 405–466.

See also "A finite presentation of the mapping class group of an oriented surface," by Gervais: https://arxiv.org/pdf/math/9811162.pdf.

• 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 '19 at 14:02
• Well to get the whole group you need all the permutations of the punctures, which you can get by throwing in half twists about curves bounding twice punctured disks. – Autumn Kent Mar 25 '19 at 15:51
• Okay, so these are $\binom{m}{2}$ additional generators, right? (maybe not all necessary) – FKranhold Mar 25 '19 at 16:06
• You can get away with a transposition (one half twist) and a cycle (which can be done using a mapping class supported on an annulus containing all the punctures). So just two more generators. – Autumn Kent Mar 25 '19 at 17:32