Relative Property (T) for the Subequivalence Relations Induced by the Action of SL₂(\Bbb Z) on \Bbb T²

2009/01/13 by Adrian Ioana, Ioana, Adrian
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Operator Algebras (math.OA) #math.DS #math.GR #math.OA

paper · pdf · doi:10.48550/arxiv.0901.1874

arxiv created 2009/01/13 · arxiv updated 2009/12/01

Abstract

Let \Cal S be the equivalence relation induced by the action SL₂(\Bbb Z)\curvearrowright (\Bbb T²,λ²), where λ² denotes the Haar measure on the 2-torus, \Bbb T². 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.

Related

• Simple and large equivalence relations
• On dense orbits in the space of subequivalence relations
• Weak containment rigidity for distal actions
• Rigidity for equivalence relations on homogeneous spaces
• Outer automorphism groups of some ergodic equivalence relations