2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191

Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups

1. 

Department of Mathematics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo, 606-8602 Kyoto, Japan

Received  October 2014 Revised  June 2015 Published  September 2015

We show the local rigidity of the standard action of the Borel subgroup of $SO_+(n,1)$ on a cocompact quotient of $SO_+(n,1)$ for $n \geq 3$.
Citation: Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191
References:
[1]

M. Asaoka, Nonhomogeneous locally free actions of the affine group, Ann. of Math., 175 (2012), 1-21. doi: 10.4007/annals.2012.175.1.1.  Google Scholar

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D. Fisher, Local rigidity of group actions: Past, present, future, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 45-97. doi: 10.1017/CBO9780511755187.003.  Google Scholar

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É. Ghys, Actions localement libres du groupe affine, Invent. Math., 82 (1985), 479-526. doi: 10.1007/BF01388867.  Google Scholar

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É. Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 163-185.  Google Scholar

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

[7]

M. Kanai, A remark on local rigidity of conformal actions on the sphere, Math. Res. Lett., 6 (1999), 675-680. doi: 10.4310/MRL.1999.v6.n6.a7.  Google Scholar

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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. doi: 10.2307/1971334.  Google Scholar

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R. de la Llave, Further rigidity properties of conformal Anosov systems, Ergodic Theory Dynam. Systems, 24 (2004), 1425-1441. doi: 10.1017/S0143385703000725.  Google Scholar

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R. S. Palais, Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc., 67 (1961), 362-364. doi: 10.1090/S0002-9904-1961-10617-4.  Google Scholar

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V. Sadovskaya, On uniformly quasiconformal Anosov systems, Math. Res. Lett., 12 (2005), 425-441. doi: 10.4310/MRL.2005.v12.n3.a12.  Google Scholar

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C. B. Yue, Smooth rigidity of rank-1 lattice actions on the sphere at infinity, Math. Res. Lett., 2 (1995), 327-338. doi: 10.4310/MRL.1995.v2.n3.a10.  Google Scholar

show all references

References:
[1]

M. Asaoka, Nonhomogeneous locally free actions of the affine group, Ann. of Math., 175 (2012), 1-21. doi: 10.4007/annals.2012.175.1.1.  Google Scholar

[2]

D. Fisher, Local rigidity of group actions: Past, present, future, in Dynamics, Ergodic Theory, and Geometry, Math. Sci. Res. Inst. Publ., 54, Cambridge Univ. Press, Cambridge, 2007, 45-97. doi: 10.1017/CBO9780511755187.003.  Google Scholar

[3]

É. Ghys, Sur les actions localement libres du group affine, Thèse de 3ème cycle, Lille, 1979. Google Scholar

[4]

É. Ghys, Actions localement libres du groupe affine, Invent. Math., 82 (1985), 479-526. doi: 10.1007/BF01388867.  Google Scholar

[5]

É. Ghys, Rigidité différentiable des groupes fuchsiens, Inst. Hautes Études Sci. Publ. Math., 78 (1993), 163-185.  Google Scholar

[6]

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

[7]

M. Kanai, A remark on local rigidity of conformal actions on the sphere, Math. Res. Lett., 6 (1999), 675-680. doi: 10.4310/MRL.1999.v6.n6.a7.  Google Scholar

[8]

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. doi: 10.2307/1971334.  Google Scholar

[9]

R. de la Llave, Further rigidity properties of conformal Anosov systems, Ergodic Theory Dynam. Systems, 24 (2004), 1425-1441. doi: 10.1017/S0143385703000725.  Google Scholar

[10]

R. S. Palais, Equivalence of nearby differentiable actions of a compact group, Bull. Amer. Math. Soc., 67 (1961), 362-364. doi: 10.1090/S0002-9904-1961-10617-4.  Google Scholar

[11]

V. Sadovskaya, On uniformly quasiconformal Anosov systems, Math. Res. Lett., 12 (2005), 425-441. doi: 10.4310/MRL.2005.v12.n3.a12.  Google Scholar

[12]

C. B. Yue, Smooth rigidity of rank-1 lattice actions on the sphere at infinity, Math. Res. Lett., 2 (1995), 327-338. doi: 10.4310/MRL.1995.v2.n3.a10.  Google Scholar

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