  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.
