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arXiv · 0901.1874

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

Adrian Ioana
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Abstract

Let $\Cal S$ be the equivalence relation induced by the action SL$_2(\Bbb Z)\curvearrowright (\Bbb T^2,λ^2)$, where $λ^2$ 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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Adrian Ioana. 2009-01-13. Relative Property (T) for the Subequivalence Relations Induced by the Action of SL$_2(\Bbb Z)$ on $\Bbb T^2$. https://arxiv.org/abs/0901.1874

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