It is known that every finitely presented group is the fundamental group of the total space of a Lefschetz fibration. In this paper, we give another proof which improves the result of Korkmaz. In addition, Korkmaz defined the genus of a finitely presented group. We also evaluate upper bounds for genera of some finitely presented groups.
"On genera of Lefschetz fibrations and finitely presented groups." Osaka J. Math. 53 (2) 351 - 376, April 2016.