Skip to main content Accessibility help

Cocycle rigidity of partially hyperbolic abelian actions with almost rank-one factors



We extend the recent progress on the cocycle rigidity of partially hyperbolic homogeneous abelian actions to the setting with rank-one factors in the universal cover. The method of proof relies on the periodic cycle functional and analysis of the cycle structure, but uses a new argument to give vanishing.



Hide All
[1] Damjanović, D.. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. J. Mod. Dyn. 1(4) (2007), 665688.
[2] Damjanović, D. and Katok, A.. Periodic cycle functionals and cocycle rigidity for certain partially hyperbolic ℝ k actions. Discrete Contin. Dyn. Syst. 13(4) (2005), 9851005.
[3] Damjanović, D. and Katok, A.. Local rigidity of partially hyperbolic actions I. KAM method and ℤ k actions on the torus. Ann. of Math. (2) 172(3) (2010), 18051858.
[4] Damjanović, D. and Katok, A.. Local rigidity of homogeneous parabolic actions: I. A model case. J. Mod. Dyn. 5(2) (2011), 203235.
[5] Damjanović, D. and Katok, A.. Local rigidity of partially hyperbolic actions. II: The geometric method and restrictions of Weyl chamber flows on SL(n, ℝ)/𝛤. Int. Math. Res. Not. IMRN 2011(19) (2011), 44054430.
[6] Gleason, A. M. and Palais, R. S.. On a class of transformation groups. Amer. J. Math. 79 (1957), 631648.
[7] Katok, A. and Spatzier, R. J.. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publ. Math. Inst. Hautes Études Sci. 79 (1994), 131156.
[8] Katok, A. and Spatzier, R. J.. Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions. Math. Res. Lett. 1(2) (1994), 193202.
[9] Katok, A. and Spatzier, R. J.. Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions. Tr. Mat. Inst. Steklova 216 (1997), 292319 (Din. Sist. i Smezhnye Vopr.).
[10] Katok, A. and Niţică, V.. Rigidity in Higher Rank Abelian Group Actions, Vol. I, Introduction and Cocycle Problem (Cambridge Tracts in Mathematics, 185) . Cambridge University Press, Cambridge, 2011.
[11] Katok, A., Niţică, V. and Török, A.. Non-abelian cohomology of abelian Anosov actions. Ergod. Th. & Dynam. Sys. 20(1) (2000), 259288.
[12] Margulis, G. A.. Discrete Subgroups of Semisimple Lie Groups (Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17) . Springer, Berlin, 1991.
[13] Mieczkowski, D.. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J. Mod. Dyn. 1(1) (2007), 6192.
[14] Ramirez, F. A.. Smooth cocycles over homogeneous dynamical systems. PhD Thesis, University of Michigan, ProQuest LLC, Ann Arbor, MI, 2010.
[15] Rosenberg, J.. Algebraic K-Theory and Its Applications (Graduate Texts in Mathematics, 147) . Springer, New York, 1994.
[16] Vinhage, K.. On the rigidity of Weyl chamber flows and Schur multipliers as topological groups. J. Mod. Dyn. 9 (2015), 2549.
[17] Vinhage, K. and Wang, Z.. Local rigidity of higher rank homogeneous abelian actions: a complete solution via the geometric method. Preprint, 2014, arXiv:1510.00848.
[18] Wang, Z. J.. Local rigidity of partially hyperbolic actions. J. Mod. Dyn. 4(2) (2010), 271327.
[19] Wang, Z. J.. New cases of differentiable rigidity for partially hyperbolic actions: symplectic groups and resonance directions. J. Mod. Dyn. 4(4) (2010), 585608.


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed