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Local rigidity of higher rank homogeneous abelian actions: a complete solution via the geometric method

Abstract

We show local and cocycle rigidity for most abelian higher-rank partially hyperbolic algebraic actions on homogeneous spaces obtained from semisimple Lie groups as well as their semidirect products. The method of proof uses a combination of geometric method and the theory of central extensions. The principal difference with previous work are the new aspects of the proof and treatment of abelian actions which are not restrictions of Weyl chamber flows. It is also the first time that partially hyperbolic twisted symmetric space examples have been treated in the literature.

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Notes

  1. For general actions, there is certain ambiguity to make this definition formal, but for homogeneous actions, the splitting is related to algebraic properties of homomorphism i See Sect. 2.

  2. Global rigidity for abelian actions has been established in certain settings as well, see for instance [10, 18, 34].

  3. Some possible examples followed from previous work, even though it was never written down. However, for these examples, strong assumptions on the toral fiber of the action is also required, which makes the these examples very special. For example, local rigidity on the quotients of the semidirect product \(SL(n,\mathbb {R})\ltimes _\rho V\) where \(\rho \) is the adjoint representation cannot be handled previously, but can be handled by our methods.

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Acknowledgements

The authors would like to thank their mutual advisor Anatole Katok for his introduction of the problem and valuable input and encouragement, as well as Aaron Brown, Ralf Spatzier and Federico Rodriguez-Hertz for helpful discussions on the subject.

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Correspondence to Zhenqi Jenny Wang.

Additional information

K. Vinhage: Based on research supported by NSF Grant DMS-1604796. Z. J. Wang: Based on research supported by NSF Grant DMS-1346876.

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Vinhage, K., Wang, Z.J. Local rigidity of higher rank homogeneous abelian actions: a complete solution via the geometric method. Geom Dedicata 200, 385–439 (2019). https://doi.org/10.1007/s10711-018-0379-5

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  • DOI: https://doi.org/10.1007/s10711-018-0379-5

Keywords

  • Local rigidity
  • Cocycle rigidity
  • Geometric method
  • Central extension
  • Algebraic K-theory

Mathematics Subject Classification

  • 37C85
  • 37C15