  Let $\Cal S$ be the equivalence relation induced by the action SL$_2(\Bbb
Z)\curvearrowright (\Bbb T^2,\lambda^2)$, where $\lambda^2$ denotes the Haar
measure on the 2-torus, $\Bbb T^2$. We prove that any ergodic subequivalence
relation $\Cal R$ of $\Cal S$ is either hyperfinite or rigid in the sense of S.
Popa ([Po06]). The proof uses an ergodic-theoretic criterion for rigidity of
countable, ergodic, probability measure preserving equivalence relations.
Moreover, we give a purely ergodic-theoretic formulation of rigidity for free,
ergodic, probability measure preserving actions of countable groups.
