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 1


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.

  • $\begingroup$ 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? $\endgroup$
    – FKranhold
    Mar 25, 2019 at 14:02
  • $\begingroup$ 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. $\endgroup$ Mar 25, 2019 at 15:51
  • $\begingroup$ Okay, so these are $\binom{m}{2}$ additional generators, right? (maybe not all necessary) $\endgroup$
    – FKranhold
    Mar 25, 2019 at 16:06
  • $\begingroup$ 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. $\endgroup$ Mar 25, 2019 at 17:32

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