We present three new inequalities tying the signature, the simplicial volume
and the Euler characteristic of surface bundles over surfaces. Two of them are
true for any surface bundle, while the third holds on a specific family of
surface bundles, namely the ones that arise through a ramified covering. These
are the main known examples of bundles with non-zero signature.Comment: 14 pages. Simplified the proof of Proposition 1.2. This is the final
version, accepted in Geometriae Dedicat
