Relative property (T) for the subequivalence relations induced by the action of SL2(Z) on T2

Let S be the equivalence relation induced by the action SL2(Z)↷(T2,λ2), where λ2 denotes the Haar measure on the 2-torus, T2. We prove that any ergodic subequivalence relation R of S is either hyperfinite or rigid in the sense of S. Popa [41]. 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.