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

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

2009-01-13

arxiv.org/abs/0901.1874v1

Mathematics

Operator Algebras

Scientific paper

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.

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