I'm studying Farb and Margalit's A primer on mapping class groups and trying to understand Wajnryb's finite presentation of Mod(S). I understand that There exists a finite presentation, but I can't understand how they got explicit Wajnryb's finite presentation. More precisely, I want to know how they knew that those 5 kinds of relation generate all relations.

It is still an open problem to find a short and simple way to derive a finite presentation for the mapping class group. The book by Farb and Margalit (in the recent preliminary version 4.00) gives a clear sketch of the known derivations, which are rather long and complicated. For the details one must consult the original papers cited in the book. There are two main steps. The first is to construct a certain 2dimensional cell complex that the mapping class group acts on and prove that this complex is simplyconnected. This was originally done in a 1980 paper of Thurston and myself. Basic properties of this complex make it clear that writing down an actual presentation is then just a matter of carrying out some long and difficult calculations, with perhaps some ingenuity to reduce the work. The second step is then to do the work and write down a presentation. Harer carried out this work shortly thereafter, and Wajnryb greatly simplified Harer's presentation in a 1983 paper. In a later 1999 paper Wajnryb gave a more elementary, selfcontained exposition of the whole story, but this is a 60page paper. The exact references are in the FarbMargalit book. I certainly hope that a more efficient approach is found someday. 


As the authors explain in the 20100422 version of their text, an explicit presentation was first obtained by Harer, using the method developed by Hatcher and Thurston. The latter defined a 2dimensional simply connected polyhedral complex on which Mod(S) acts :
A general recipe then says how to obtain a finite presentation of the whole group Mod(S), which Harer was the first to go through in all cases, obtaining a "somewhat unwieldy presentation". Then Wajnryb managed to simplify Harer's presentation. Personally, I prefer Makoto Matsumoto's beautiful presentation (obtained by simplifying Wajnryb's one, with the aid of a computer), where relations added to the braid ones are naturally indexed by some Dynkin diagrams (ADE again!) BTW, here's the link to Farb and Margalit's book. 


I recently ran across this paper, which might be the shortest selfcontained derivation of the Gervais presentation out there: 

