Construction of Lefschetz fibrations on 4-manifolds with a given fundamental group

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.