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First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

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Abstract

This is the first in a series of papers exploring rigidity properties of hyperbolic actions ofZ k orR k fork ≥ 2. We show that for all known irreducible examples, the cohomology of smooth cocycles over these actions is trivial. We also obtain similar Hölder and C1 results via a generalization of the Livshitz theorem for Anosov flows. As a consequence, there are only trivial smooth or Hölder time changes for these actions (up to an automorphism). Furthermore, small perturbations of these actions are Hölder conjugate and preserve a smooth volume.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, 16802, State College, PA

    Anatole Katok

  2. Department of Mathematics, University of Michigan, 48109, Ann Arbor, MI

    Ralf J. Spatzier

Authors
  1. Anatole Katok
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  2. Ralf J. Spatzier
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Additional information

Partially supported by NSF grant DMS 9017995.

Partially supported by the NSF, AMS Centennial Fellow.

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Katok, A., Spatzier, R.J. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publications Mathématiques de L’Institut des Hautes Scientifiques 79, 131–156 (1994). https://doi.org/10.1007/BF02698888

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  • DOI: https://doi.org/10.1007/BF02698888

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