Geometric monodromy and the hyperbolic disc

Symplectic four-manifolds give rise to Lefschetz fibrations, which are dete rmined by monodromy representations of free groups in mapping class groups. We study the topology of Lefschetz fibrations by analysing the action of t he monodromy on the universal cover of a smooth fibre and give a new and si mple proof that Lefschetz fibrations arising from Donaldson's construction via pencils of sections never decompose as non-trivial fibre sums; in parti cular not all Lefschetz fibrations are fibre sums of holomorphic Lefschetz fibrations. We also show that there can never be isotopy classes of simple closed curve invariant under the monodromy and as a corollary we give a sym plectic analogue of Manin's theorem, showing that Lefschetz fibrations admi t at most finitely many homotopy classes of geometric section.