Every finitely presented group is the fundamental group of the total space of a Lefschetz fibration over 2-sphere. This follows from results of Gompf and Donaldson, and was also proved by Amoros-Bogomolov-Katzarkov-Pantev. I will give another proof by providing the monodromy explicitly. Then I will discuss an invariant of finitely presented groups obtained this way.
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/node/249