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

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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

Abstract
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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.
[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.
[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.
[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.
[5]	Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: 10.2307/2373853.
[6]	A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55-58.
[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.
[8]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[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.
[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).
[11]	A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191-210.
[12]	A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions," Cambridge University Press, to appear.
[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.
[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.
[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.
[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.
[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.
[18]	H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–-62.
[19]	C. Moore, Group extensions of p-adic and adelic linear groups, Inst. Hautes Etudes Sci. Publ. Math., No. 35, (1968), 157-222.
[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.
[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.
[22]	J. R. Silvester, "Introduction to Algebraic K-Theory," Chapman and Hall Mathematics Series. Chapman & Hall, London-New York, 1981.
[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.
[24]	R. Steinberg, "Lecture Notes on Chevalley Groups,'' Yale Univ., 1967.
[25]	Zhenqi Wang, Local rigidity of partially hyperbolic actions, Journal of Modern Dynamics, 4 (2010), 271-327. doi: 10.3934/jmd.2010.4.271.

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.
[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.
[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.
[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.
[5]	Vinay V. Deodhar, On central extensions of rational points of algebraic groups, Amer. J. Math., 100 (1978), 303-386. doi: 10.2307/2373853.
[6]	A. J. Hahn and O. T. O'Meara, The classical groups and K-theory, Springer Verlag, Berlin, 1980, 55-58.
[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.
[8]	M. Hirsch, C. Pugh and M. Shub, "Invariant Manifolds,'' Lecture Notes in Mathematics, 583, Springer-Verlag, Berlin-New York, 1977.
[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.
[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).
[11]	A. Katok and A. Kononenko, Cocycle stability for partially hyperbolic systems, Math. Res. Letters, 3 (1996), 191-210.
[12]	A. Katok and V. Nitica, "Differentiable Rigidity of Higher-Rank Abelian Group Actions," Cambridge University Press, to appear.
[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.
[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.
[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.
[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.
[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.
[18]	H. Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4), 2 (1969), 1–-62.
[19]	C. Moore, Group extensions of p-adic and adelic linear groups, Inst. Hautes Etudes Sci. Publ. Math., No. 35, (1968), 157-222.
[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.
[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.
[22]	J. R. Silvester, "Introduction to Algebraic K-Theory," Chapman and Hall Mathematics Series. Chapman & Hall, London-New York, 1981.
[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.
[24]	R. Steinberg, "Lecture Notes on Chevalley Groups,'' Yale Univ., 1967.
[25]	Zhenqi Wang, Local rigidity of partially hyperbolic actions, Journal of Modern Dynamics, 4 (2010), 271-327. doi: 10.3934/jmd.2010.4.271.

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