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 (, c₁^2) = (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.