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October  2010, 4(4): 585-608. doi: 10.3934/jmd.2010.4.585

New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions

Zhenqi Jenny Wang 1,

1. 

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

Received  January 2010 Revised  September 2010 Published  January 2011

We prove the local differentiable rigidity of generic partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from split symplectic Lie groups. We also give examples of rigidity for nongeneric actions on compact homogeneous spaces obtained from SL$(2n,\RR)$ or SL$(2n,\CC)$. The conclusions are based on the geometric approach by Katok--Damjanovic and a progress towards computations of the generating relations in these groups.
Keywords: Actions of higher rank abelian groups, generators and relations in classical Lie groups, rigidity, cocycles, partial hyperbolicity., Steinberg symbols.
Mathematics Subject Classification: Primary: 37D; Secondary: 19C, 22.
Citation: Zhenqi Jenny Wang. New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions. Journal of Modern Dynamics, 2010, 4 (4) : 585-608. doi: 10.3934/jmd.2010.4.585
References:
[1]

M. Brin and Y. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar

[2]

D. Damjanović and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\RR^k$ actions, Discr. Cont. Dyn. Syst., 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985.  Google Scholar

[3]

D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on on $\SL(n,\RR)/\Gamma$, Int. Math. Res. Notes, 2010, to appear. Google Scholar

[4]

D. Damjanovi$\acutec$, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Modern Dyn., 1 (2007), 665-688.  Google Scholar

[5]

Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: 10.2307/2373853.  Google Scholar

[6]

A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55-58.  Google Scholar

[7]

S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces," Corrected reprint of the 1978 original, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[8]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

B. Kalinin and R. Spatzier, On the classification of Cartan actions, Geom. Funct. Anal., 17 (2007), 468-490. doi: 10.1007/s00039-007-0602-2.  Google Scholar

[10]

B. Kalinin and A. Katok, Invariant measures for actions of higher-rank abelian groups, Smooth Ergodic Theory and its applications (Seattle,WA, 1999), 593-637, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001).  Google Scholar

[11]

A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191-210.  Google Scholar

[12]

A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions,", Cambridge University Press, ().   Google Scholar

[13]

A. Katok and R. Spatzier, First cohomology of Anosov actions of higher-rank abelian groups and applications to rigidity, Inst. Hautes čtudes Sci. Publ. Math. No. 79, (1994), 131-156.  Google Scholar

[14]

A. Katok and R. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Letters, 1 (1994), 193-202.  Google Scholar

[15]

A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher-rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova, 216 (1997), Din. Sist. i Smezhnye Vopr., 292-319; translation in Proc. Steklov Inst. Math., 1997, 287-314.  Google Scholar

[16]

G. A.Margulis, "Discrete Subgroups Of Semisimple Lie Groups,'' Ergebnisse derMathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[17]

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: 10.1017/S0143385701001109.  Google Scholar

[18]

H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–-62.  Google Scholar

[19]

C. Moore, Group extensions of p-adic and adelic linear groups, Inst. Hautes Etudes Sci. Publ. Math., No. 35, (1968), 157-222.  Google Scholar

[20]

J. Milnor, "Introduction to Algebraic K-theory,'' Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.  Google Scholar

[21]

Y. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity,'' Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[22]

J. R. Silvester, "Introduction to Algebraic K-Theory," Chapman and Hall Mathematics Series. Chapman & Hall, London-New York, 1981.  Google Scholar

[23]

R. Steinberg, Générateurs, relations et revêtements de groupes algébriques, (French) 1962 Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) 113-127 Librairie Universitaire, Louvain; Gauthier-Villars, Paris.  Google Scholar

[24]

R. Steinberg, "Lecture Notes on Chevalley Groups,'' Yale Univ., 1967.  Google Scholar

[25]

Zhenqi Wang, Local rigidity of partially hyperbolic actions, Journal of Modern Dynamics, 4 (2010), 271-327. doi: 10.3934/jmd.2010.4.271.  Google Scholar

show all references

References:
[1]

M. Brin and Y. Pesin, Partially hyperbolic dynamical systems, (Russian) Izv. Akad. Nauk SSSR Ser. Mat., 38 (1974), 170-212.  Google Scholar

[2]

D. Damjanović and A. Katok, Periodic cycle functionals and Cocycle rigidity for certain partially hyperbolic $\RR^k$ actions, Discr. Cont. Dyn. Syst., 13 (2005), 985-1005. doi: 10.3934/dcds.2005.13.985.  Google Scholar

[3]

D. Damjanović and A. Katok, Local rigidity of partially hyperbolic actions. II. The geometric method and restrictions of Weyl Chamber flows on on $\SL(n,\RR)/\Gamma$, Int. Math. Res. Notes, 2010, to appear. Google Scholar

[4]

D. Damjanovi$\acutec$, Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions, J. Modern Dyn., 1 (2007), 665-688.  Google Scholar

[5]

Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: 10.2307/2373853.  Google Scholar

[6]

A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55-58.  Google Scholar

[7]

S. Helgason, "Differential Geometry, Lie Groups, and Symmetric Spaces," Corrected reprint of the 1978 original, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001.  Google Scholar

[8]

M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[9]

B. Kalinin and R. Spatzier, On the classification of Cartan actions, Geom. Funct. Anal., 17 (2007), 468-490. doi: 10.1007/s00039-007-0602-2.  Google Scholar

[10]

B. Kalinin and A. Katok, Invariant measures for actions of higher-rank abelian groups, Smooth Ergodic Theory and its applications (Seattle,WA, 1999), 593-637, Proc. Sympos. Pure Math., 69, Amer. Math. Soc., Providence, RI, (2001).  Google Scholar

[11]

A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191-210.  Google Scholar

[12]

A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions,", Cambridge University Press, ().   Google Scholar

[13]

A. Katok and R. Spatzier, First cohomology of Anosov actions of higher-rank abelian groups and applications to rigidity, Inst. Hautes čtudes Sci. Publ. Math. No. 79, (1994), 131-156.  Google Scholar

[14]

A. Katok and R. Spatzier, Subelliptic estimates of polynomial differential operators and applications to rigidity of abelian actions, Math. Res. Letters, 1 (1994), 193-202.  Google Scholar

[15]

A. Katok and R. Spatzier, Differential rigidity of Anosov actions of higher-rank abelian groups and algebraic lattice actions, Tr. Mat. Inst. Steklova, 216 (1997), Din. Sist. i Smezhnye Vopr., 292-319; translation in Proc. Steklov Inst. Math., 1997, 287-314.  Google Scholar

[16]

G. A.Margulis, "Discrete Subgroups Of Semisimple Lie Groups,'' Ergebnisse derMathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 17, Springer-Verlag, Berlin, 1991.  Google Scholar

[17]

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: 10.1017/S0143385701001109.  Google Scholar

[18]

H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–-62.  Google Scholar

[19]

C. Moore, Group extensions of p-adic and adelic linear groups, Inst. Hautes Etudes Sci. Publ. Math., No. 35, (1968), 157-222.  Google Scholar

[20]

J. Milnor, "Introduction to Algebraic K-theory,'' Annals of Mathematics Studies, No. 72. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971.  Google Scholar

[21]

Y. Pesin, "Lectures on Partial Hyperbolicity and Stable Ergodicity,'' Zürich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zürich, 2004. doi: 10.4171/003.  Google Scholar

[22]

J. R. Silvester, "Introduction to Algebraic K-Theory," Chapman and Hall Mathematics Series. Chapman & Hall, London-New York, 1981.  Google Scholar

[23]

R. Steinberg, Générateurs, relations et revêtements de groupes algébriques, (French) 1962 Colloq. Théorie des Groupes Algébriques (Bruxelles, 1962) 113-127 Librairie Universitaire, Louvain; Gauthier-Villars, Paris.  Google Scholar

[24]

R. Steinberg, "Lecture Notes on Chevalley Groups,'' Yale Univ., 1967.  Google Scholar

[25]

Zhenqi Wang, Local rigidity of partially hyperbolic actions, Journal of Modern Dynamics, 4 (2010), 271-327. doi: 10.3934/jmd.2010.4.271.  Google Scholar

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