Local rigidity of partially hyperbolic actions

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2010, 17: 68-79. doi: 10.3934/era.2010.17.68

Local rigidity of partially hyperbolic actions

Zhenqi Jenny Wang ^1,

1.	Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States

Received February 2010 Revised July 2010 Published September 2010

Abstract
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We prove the local differentiable rigidity of partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. The conclusions are based on geometric Katok-Damjanovic way and progress towards computations of the Schur multipliers of these non-split groups.

Keywords: Schur multiplier., Local rigidity, group actions.

Mathematics Subject Classification: 37C8.

Citation: Zhenqi Jenny Wang. Local rigidity of partially hyperbolic actions. Electronic Research Announcements, 2010, 17: 68-79. doi: 10.3934/era.2010.17.68

References:

[1]	M. Brin, Y. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.
[2]	D. Damjanovic and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\RR^k$ actions, Discr. Cont. Dyn.Syst., 13 (2005), 985-1005.
[3]	D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\RR)/\Gamma$,, , ().
[4]	D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. I. KAM method and $\ZZ^k$ actions on the torus, Annals of Math, 2010, to appear.
[5]	D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Modern Dyn., 1 (2007), 665-688.
[6]	Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: doi:10.2307/2373853.
[7]	A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55-58.
[8]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds. Lecture Notes in Mathematics," 583, Springer Verlag, Berlin, 1977.
[9]	A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 216 (1997), 287-314.
[10]	G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups," Springer-Verlag, 1991.
[11]	G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems, 21 (2001), 121-164. doi: doi:10.1017/S0143385701001109.
[12]	R. Steinberg, Generateurs, relations et revetements de groupes algebriques, Colloque de Bruxelles, 1962, 113-127.

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References:

[1]	M. Brin, Y. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.
[2]	D. Damjanovic and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\RR^k$ actions, Discr. Cont. Dyn.Syst., 13 (2005), 985-1005.
[3]	D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on $SL(n,\RR)/\Gamma$,, , ().
[4]	D. Damjanovic and A. Katok, Local rigidity of partially hyperbolic actions. I. KAM method and $\ZZ^k$ actions on the torus, Annals of Math, 2010, to appear.
[5]	D. Damjanovic, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Modern Dyn., 1 (2007), 665-688.
[6]	Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: doi:10.2307/2373853.
[7]	A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55-58.
[8]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds. Lecture Notes in Mathematics," 583, Springer Verlag, Berlin, 1977.
[9]	A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher rank abelian groups and algebraic lattice actions, Proc. Steklov Inst. Math., 216 (1997), 287-314.
[10]	G. A. Margulis, "Discrete Subgroups of Semisimple Lie Groups," Springer-Verlag, 1991.
[11]	G. A. Margulis and N. Qian, Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices, Ergodic Theory Dynam. Systems, 21 (2001), 121-164. doi: doi:10.1017/S0143385701001109.
[12]	R. Steinberg, Generateurs, relations et revetements de groupes algebriques, Colloque de Bruxelles, 1962, 113-127.

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