Geography of symplectic 4-manifolds admitting Lefschetz fibrations and their indecomposability

In this paper, we show that for a given finitely presented group G, there exist integers hG ≥ 0 and nG ≥ 4 such that for all h ≥ hG and n ≥ nG, and for all 0 ≤ i ≤ 2n - 2, there exists a genus-$(2h + n - 1)$ Lefschetz fibration on a minimal symplectic 4-manifold with (χ, c₁²) = (n, i) whose fundamental group is isomorphic to G. We also prove that such a fibration cannot be decomposed as a fiber sum for 1 ≤ i ≤ 2n - 2 if $h > (5n - 3)/2$. In addition, we give a relation among the genus of the base space of a ruled surface admitting a Lefschetz fibration, the number of blow-ups and the genus of the Lefschetz fibration.