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

Abstract

Let $\Cal S$ be the equivalence relation induced by the action SL${}_2(\mathbb{Z})\curvearrowright ({\mathbb{T}}^2,{\lambda }^2)$, where ${\lambda }^2$ denotes the Haar measure on the 2-torus, ${\mathbb{T}}^2$. 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.