A genus-$g$ Lefschetsz fibration on a closed orientable smooth $4$-manifold $X$ is a smooth map from $X$ onto a surface admitting certain singular fibers such that regular fibers are closed orientable surfaces of genus $g$. By results of Gompf and Donaldson, the total space of every Lefshetz fibration is symplectic for $g\geq 2$, and conversely, every symplectic $4$-manifold admits a Lefschetz fibration perhaps after bowing up. In this way, symlectic $4$-manifolds can be viewed as certain relations between Dehn twists in the mapping class group.
Let $\Gamma$ be a finitely presented group. It is known that $\Gamma$ is the fundamental group of a closed symplectic 4-manifold, and hence, that of the total space of a Lefschetz fibration. Another construction is due to Amoros, Bogomolov, Katzarkov and Pantev. In this talk I will describe another method of the construction of a Lefschetz fibration with the fundamental group $\Gamma$. It turns out that the monodromy can be given explicitely. From this, I will derive an invariant of finitely presented group.