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Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions

  • Adrian Ioana EMAIL logo
Published/Copyright: May 28, 2015
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2017 Issue 733

Abstract

We study equivalence relations that arise from translation actions Γ↷G which are associated to dense embeddings Γ<G of countable groups into second countable locally compact groups. Assuming that G is simply connected and the action Γ↷G is strongly ergodic, we prove that Γ↷G is orbit equivalent to another such translation action Λ↷H if and only if there exists an isomorphism δ:G→H such that δ⁢(Γ)=Λ. If G is moreover a real algebraic group, then we establish analogous rigidity results for the translation actions of Γ on homogeneous spaces of the form G/Σ, where Σ<G is either a discrete or an algebraic subgroup. We also prove that if G is simply connected and the action Γ↷G has property (T), then any cocycle w:Γ×G→Λ with values in a countable group Λ is cohomologous to a homomorphism δ:Γ→Λ. As a consequence, we deduce that the action Γ↷G is orbit equivalent superrigid: any free nonsingular action Λ↷Y which is orbit equivalent to Γ↷G, is necessarily conjugate to an induction of Γ↷G.

Funding source: National Science Foundation

Award Identifier / Grant number: DMS #1161047

Award Identifier / Grant number: DMS #1253402

Funding statement: The author was partially supported by NSF Grant DMS #1161047, NSF Career Grant DMS #1253402, and a Sloan Foundation Fellowship.

Acknowledgements

I would like to thank Alireza Salehi-Golsefidy for helpful discussions on algebraic groups and in particular for pointing out Remark 10.2.

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Received: 2014-9-13
Published Online: 2015-5-28
Published in Print: 2017-12-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Lagrangian F-stability of closed Lagrangian self-shrinkers
  3. Equidistribution in supersingular Hecke orbits
  4. Yau–Tian–Donaldson correspondence for K-semistable Fano manifolds
  5. Convergence to equilibrium of gradient flows defined on planar curves
  6. Potential theory for manifolds with boundary and applications to controlled mean curvature graphs
  7. Image of the braid groups inside the finite Iwahori–Hecke algebras
  8. Wall-crossing in genus zero Landau–Ginzburg theory
  9. Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
  10. On the boundedness of the denominators in the Zariski decomposition on surfaces
https://doi.org/10.1515/crelle-2014-0155

Articles in the same Issue

  1. Frontmatter
  2. Lagrangian F-stability of closed Lagrangian self-shrinkers
  3. Equidistribution in supersingular Hecke orbits
  4. Yau–Tian–Donaldson correspondence for K-semistable Fano manifolds
  5. Convergence to equilibrium of gradient flows defined on planar curves
  6. Potential theory for manifolds with boundary and applications to controlled mean curvature graphs
  7. Image of the braid groups inside the finite Iwahori–Hecke algebras
  8. Wall-crossing in genus zero Landau–Ginzburg theory
  9. Strong ergodicity, property (T), and orbit equivalence rigidity for translation actions
  10. On the boundedness of the denominators in the Zariski decomposition on surfaces
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