  We show vanishing results about the infimum of the topological entropy of the
geodesic flow of homogeneous smooth four manifolds. We prove that any closed
oriented geometric four manifold has zero minimal entropy if and only if it has
zero simplicial volume. We also show that if a four manifold M admits a
geometric decomposition, in the sense of Thurston, and does not have geometric
pieces modelled on hyperbolic four-space, the complex hyperbolic plane or the
product of two hyperbolic planes, then M admits an F-structure. It follows that
M has zero minimal entropy and collapses with curvature bounded from below. We
then analyse whether or not M admits a metric whose topological entropy
coincides with the minimal entropy of M and provide new examples of manifolds
for which the minimal entropy problem cannot be solved.
