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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.

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3 Answers 3

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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 2-dimensional cell complex that the mapping class group acts on and prove that this complex is simply-connected. 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, self-contained exposition of the whole story, but this is a 60-page paper. The exact references are in the Farb-Margalit book. I certainly hope that a more efficient approach is found someday.

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

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@Dan : The link seems to be broken. Which paper is this? – Andy Putman Feb 13 '11 at 17:45
I'm guessing it's Benvenuti's paper? If so, I still find it pretty painful. – Daniel Moskovich Feb 13 '11 at 19:00
@Theo : I fixed the link. – Andy Putman Feb 13 '11 at 19:03
@Andy: Thank you! – Theo Buehler Feb 13 '11 at 19:25

As the authors explain in the 2010-04-22 version of their text, an explicit presentation was first obtained by Harer, using the method developed by Hatcher and Thurston. The latter defined a 2-dimensional simply connected polyhedral complex on which Mod(S) acts :

  1. cocompactly (i.e. finitely many orbits of 0,1, and 2 faces)
  2. with explicitly finitely presented vertex stabilizers (close to braid groups)
  3. with explicitly finitely generated edge stabilizers

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.

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