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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.