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Let $\Cal S$ be the equivalence relation induced by the action SL$_2(\BbbZ)\curvearrowright (\Bbb T^2,\lambda^2)$, where $\lambda^2$ denotes the Haarmeasure on the 2-torus, $\Bbb T^2$. We prove that any ergodic subequivalencerelation $\Cal R$ of $\Cal S$ is either hyperfinite or rigid in the sense of S.Popa ([Po06]). The proof uses an ergodic-theoretic criterion for rigidity ofcountable, 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.

Relative Property (T) for the Subequivalence Relations Induced by the Action of SL$_2(\Bbb Z)$ on $\Bbb T^2$