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Parameter rigid actions of simply connected nilpotent Lie groups

Published online by Cambridge University Press:  22 August 2012

HIROKAZU MARUHASHI
HIROKAZU MARUHASHI*
Affiliation:
Department of Mathematics, Kyoto University, Japan (email: h-maruha@math.kyoto-u.ac.jp)
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Abstract

We show that for a locally free $C^{\infty }$-action of a connected and simply connected nilpotent Lie group on a compact manifold, if every real-valued cocycle is cohomologous to a constant cocycle, then the action is parameter rigid. The converse is true if the action has a dense orbit. Using this, we construct parameter rigid actions of simply connected nilpotent Lie groups whose Lie algebras admit rational structures with graduations. This generalizes the results of dos Santos [Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys.27(2007), 1719–1735] concerning the Heisenberg groups.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 6 , December 2013 , pp. 1864 - 1875
Copyright
Copyright © 2012 Cambridge University Press 

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References

[1]Asaoka, M.. Deformation of locally free actions and the leafwise cohomology. arXiv:1012.2946.Google Scholar
[2]Corwin, L. and Greenleaf, F. P.. Representations of Nilpotent Lie Groups and Their Applications. Part 1:Basic Theory and Examples (Cambridge Studies in Advanced Mathematics, 18). Cambridge University Press, Cambridge, 1990.Google Scholar
[3]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), 131–156.CrossRefGoogle Scholar
[4]Matsumoto, S. and Mitsumatsu, Y.. Leafwise cohomology and rigidity of certain Lie group actions. Ergod. Th. & Dynam. Sys. 23 (2003), 1839–1866.Google Scholar
[5]Mieczkowski, D.. The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory. J. Mod. Dyn. 1 (2007), 61–92.Google Scholar
[6]Pereira, M. S. and dos Santos, N. M.. On the cohomology of foliated bundles. Proyecciones 21(2) (2002), 175–197.Google Scholar
[7]Ramírez, F. A.. Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups. J. Mod. Dyn. 3 (2009), 335–357.CrossRefGoogle Scholar
[8]dos Santos, N. M.. Parameter rigid actions of the Heisenberg groups. Ergod. Th. & Dynam. Sys. 27 (2007), 1719–1735.Google Scholar