Lefschetz fibrations and an invariant of finitely presented groups

Every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. This follows from results of Gompf and Donaldson, and was also proved by Amoros-Bogomolov-Katzarkov-Pantev. We give another proof by providing the monodromy explicitly. We then define the genus of a finitely presented group $\Gamma$ to be the minimal genus of a Lefschetz fibration with fundamental group $\Gamma$. We also give some estimates of the genus of certain groups.