Skip to main content
SpringerLink
Log in

Higher cohomology for Anosov actions on certain homogeneous spaces

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We study the smooth untwisted cohomology with real coefficients for the action on [SL(2,ℝ)×…×SL(2,ℝ)]/Γ by the subgroup of diagonal matrices, where Γ is an irreducible lattice. We show that in the top degree, the obstructions to solving the coboundary equation come from distributions that are invariant under the action. We also show that in intermediate degrees, the cohomology trivializes. It has been conjectured by A. Katok and S. Katok that, analogously to Livšic’s theorem for Anosov flows for a standard partially hyperbolic ℝd - or ℤd - action, the obstructions to solving the top-degree coboundary equation are given by periodic orbits, and also that the intermediate cohomology trivializes, as it is known to do in the first degree by work of Katok and Spatzier. Katok and Katok proved their conjecture for abelian groups of toral automorphisms. Our results verify the “intermediate cohomology” part of the conjecture for diagonal subgroup actions on SL(2,ℝ)d /Γ and are a step in the direction of the “top-degree cohomology” part.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Laurent Clozel, Démonstration de la conjecture τ, Invent. Math. 151 (2003), 297–328.

    Article  MathSciNet  MATH  Google Scholar 

  2. Danijela Damjanović and Anatole Katok, Local rigidity of partially hyperbolic actions I. KAM method and ℤk actions on the torus, Ann. of Math. (2) 172 (2010), 1805–1858.

    Article  MathSciNet  MATH  Google Scholar 

  3. Danijela Damjanović and Anatole Katok, Local rigidity of homogeneous parabolic actions: I. A model case, J. Mod. Dyn. 5 (2011), 203–235.

    Article  MathSciNet  MATH  Google Scholar 

  4. R. de la Llave, J. M. Marco, and R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2) 123 (1986), 537–611.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. Flaminio and G. Forni, Invariant distributions and time averages for horocycle flows, Duke Math. J. 119 (2003), 465–526.

    Article  MathSciNet  MATH  Google Scholar 

  6. T. Foth and S. Katok, Spanning sets for automorphic forms and dynamics of the frame flow on complex hyperbolic spaces, Ergodic Theory Dynam. Systems 21 (2001), 1071–1099.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Guillemin and D. Kazhdan, On the cohomology of certain dynamical systems, Topology 19 (1980), 291–299.

    Article  MathSciNet  MATH  Google Scholar 

  8. R. Howe and E-C. Tan, Nonabelian Harmonic Analysis, Springer-Verlag, New York, 1992.

    Google Scholar 

  9. A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms, Ergodic Theory Dynam. Systems 15 (1995), 569–592.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Katok and S. Katok, Higher cohomology for abelian groups of toral automorphisms. II. The partially hyperbolic case, and corrigendum, Ergodic Theory Dynam. Systems 25 (2005), 1909–1917.

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Katok and R. J. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity, Inst. Hautes Études Sci. Publ.Math. 79 (1994), 131–156.

    Article  MathSciNet  MATH  Google Scholar 

  12. A. Katok and Z. J. Wang, Local rigidity of partially hyperbolic actions: solution of the general problem via KAM method, preprint, 2011.

  13. S. Katok, Closed geodesics, periods and arithmetic of modular forms, Invent. Math. 80 (1985), 469–480.

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Y. Kleinbock and G. A. Margulis, Logarithm laws for flows on homogeneous spaces, Invent. Math. 138 (1999), 451–494.

    Article  MathSciNet  MATH  Google Scholar 

  15. S. Lang, SL 2(R), Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975.

    MATH  Google Scholar 

  16. A. N. Livšic, Cohomology of dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1296–1320.

    MathSciNet  Google Scholar 

  17. F. I. Mautner, Unitary representations of locally compact groups. I, Ann. of Math.(2) 51 (1950), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  18. F. I. Mautner, Unitary representations of locally compact groups. II, Ann. of Math.(2) 52 (1950), 528–556.

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Mieczkowski, The cohomological equation and representation theory, Ph.D. thesis, The Pennsylvania State University, 2006.

  20. D. Mieczkowski, The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory, J. Mod. Dyn. 1 (2007), 61–92.

    Article  MathSciNet  MATH  Google Scholar 

  21. F. A. Ramírez, Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups, J. Mod. Dyn. 3 (2009), 335–357.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. E. Taylor, Noncommutative Harmonic Analysis, American Mathematical Society, Providence, RI, 1986.

    Book  MATH  Google Scholar 

  23. W. A. Veech, Periodic points and invariant pseudomeasures for toral endomorphisms, Ergodic Theory Dynam. Systems 6 (1986)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of York, York, YO10 5DD, UK

    F. A. Ramírez

Authors
  1. F. A. Ramírez
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. A. Ramírez.

Additional information

The author was partially supported by NSF RTG number DMS-0602191

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ramírez, F.A. Higher cohomology for Anosov actions on certain homogeneous spaces. JAMA 121, 177–222 (2013). https://doi.org/10.1007/s11854-013-0032-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-013-0032-z

Keywords

Search

Navigation