  We prove that the group of Hamiltonian automorphisms of a symplectic
4-manifold contains only finitely many conjugacy classes of maximal compact
tori with respect to the action of the full symplectomorphism group. We also
extend to rational and ruled manifolds a result of Kedra which asserts that, if
$M$ is a simply connected symplectic 4-manifold with $b_{2}\geq 3$, and if
$\widetilde{M}_{\delta}$ denotes a blow-up of $M$ of small enough capacity
$\delta$, then the rational cohomology algebra of the Hamiltonian group of
$\widetilde{M}_{\delta})$ is not finitely generated. Both results are based on
the fact that in a symplectic 4-manifold endowed with any tamed almost
structure $J$, exceptional classes of minimal symplectic area are
$J$-indecomposable. Some applications and examples are given.
