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 take 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.

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.

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 explicitely finitely presented vertex stabilizers (close to braid groups)
3. with explicitely 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 take 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!)

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 take 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.