Let ($M$, \omega) be a four-dimensional compact connected symplectic manifold. We
prove that ($M$, \omega) admits only finitely many inequivalent Hamiltonian
effective 2-torus actions. Consequently, if $M$ is simply connected, the number
of conjugacy classes of 2-tori in the symplectomorphism group Sympl($M$, \omega)
is finite. Our proof is “soft”. The proof uses the fact that if a symplectic
four-manifold admits a toric action, then the restriction of the period map to
the set of exceptional homology classes is proper. In an appendix, we present
the Gromov–McDuff properness result for a general compact symplectic
four-manifold.