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Deformations of group actions


Author: David Fisher
Journal: Trans. Amer. Math. Soc. 360 (2008), 491-505
MSC (2000): Primary 37C85
DOI: https://doi.org/10.1090/S0002-9947-07-04372-3
Published electronically: July 20, 2007
MathSciNet review: 2342012
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Abstract: Let $ G$ be a non-compact real algebraic group and $ \Gamma<G$ a lattice. One purpose of this paper is to show that there is a smooth, volume preserving, mixing action of $ G$ or $ \Gamma$ on a compact manifold which admits a smooth deformation. In fact, we prove a stronger statement by exhibiting large finite dimensional spaces of deformations. We also describe some other, rather special, deformations when $ G=SO(1,n)$.


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  • 1. Louis Auslander, On radicals of discrete subgroups of Lie groups. Amer. J. Math. 85 1963 145-150. MR 0152607 (27:2583)
  • 2. E.J.Benveniste, Exotic Actions of Semisimple Groups and their Deformations, unpublished manuscript and chapter of doctoral dissertation, University of Chicago, 1996.
  • 3. E.J.Benveniste, Rigidity of isometric lattice actions on compact Riemannian manifolds, GAFA 10 (2000) 516-542. MR 1779610 (2001j:37055)
  • 4. E.J.Benveniste and D.Fisher, Nonexistence of invariant rigid structures and invariant almost rigid structure, Comm. Anal. Geom. 13 (2005) 89-111. MR 2154667 (2006f:53056)
  • 5. D.Fisher, Bending group actions and cohomology of arithmetic groups, in preparation.
  • 6. D.Fisher and G.A.Margulis, Local rigidity of affine actions of higher rank groups and lattices, preprint.
  • 7. D.Fisher and G.A.Margulis, Almost isometries, Property $ (T)$ and local rigidity, Invent. Math., 162 (2005) 19-80. MR 2198325 (2006m:53063)
  • 8. D.Fisher and G.A.Margulis, Local rigidity for cocycles,in Surv. Diff. Geom. Vol VIII, refereed volume in honor of Calabi, Lawson, Siu and Uhlenbeck , editor: S.T. Yau, 45 pages, 2003. MR 2039990 (2004m:22032)
  • 9. D. Fisher, and K. Whyte, Continuous quotients for lattice actions on compact spaces, Geom. Dedicata, 87 (2001), no. 1-3, 181-189. MR 1866848 (2002j:57070)
  • 10. M.W. Hirsch, C.C. Pugh, and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics 583 , Springer-Verlag, New York, 1977. MR 0501173 (58:18595)
  • 11. T.J.Hitchman, Rigidity theorems for large dynamical systems with hyperbolic behavior. Ph.D. Thesis, University of Michigan, 2003.
  • 12. Dennis Johnson and John J. Millson, Deformation spaces associated to compact hyperbolic manifolds. Discrete groups in geometry and analysis (New Haven, Conn., 1984), 48-106, Progr. Math., 67, Birkhäuser Boston, Boston, MA, 1987. MR 900823 (88j:22010)
  • 13. A. Katok and J. Lewis, Global rigidity results for lattice actions on tori and new examples of volume-preserving actions. Israel J. Math. 93 (1996), 253-280. MR 1380646 (96k:22021)
  • 14. G.A. Margulis, Discrete subgroups of semisimple Lie groups, Springer-Verlag, New York, 1991. MR 1090825 (92h:22021)
  • 15. S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold. Ann. of Math. 40 (1939), no. 2, 400-416. MR 1503467
  • 16. Dave Witte Morris, Introduction to Arithmetic Groups, preprint, available at http://people.uleth.ca/ dave.morris/ or via math arxiv.
  • 17. Richard S. Palais, Equivalence of nearby differentiable actions of a compact group. Bull. Amer. Math. Soc. 67 (1961) 362-364. MR 0130321 (24:A185)
  • 18. V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, New York, 1994. MR 1278263 (95b:11039)
  • 19. M. Ratner, On Raghunathan's measure conjectures, Ann. of Math. 134 no. 3 (1991) 545-607. MR 1135878 (93a:22009)
  • 20. N. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, Proceedings of the International Colloquium on Lie Groups and Ergodic Theory Mumbai 1996, edited by S.G. Dani.
  • 21. D. Witte, Measurable Quotients of Unipotent Translations on Homogeneous Spaces, Trans. Amer. Math. Soc. 354 no. 2 (1994) 577-594. MR 1181187 (95a:22005)
  • 22. D.Witte, Cocycle superrigidity for ergodic actions of non-semisimple Lie groups. Lie groups and ergodic theory (Mumbai, 1996), 367-386, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998. MR 1699372 (2000i:22008)
  • 23. R.J. Zimmer, Ergodic Theory and semisimple Groups, Birkhäuser, Boston, 1984. MR 776417 (86j:22014)
  • 24. R.J. Zimmer, Actions of semisimple groups and discrete subgroups. Proc. Internat. Cong. Math. (Berkeley, 1986) 1247-1258 (1987) MR 934329 (89j:22024)
  • 25. R.J. Zimmer, Lattices in semisimple groups and invariant geometric structures on compact manifolds. In: Discrete groups in geometry and analysis, R. Howe, editor. Prog. in Math 67, 152-210. Boston: Birkhauser 1987 MR 900826 (88i:22025)

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Additional Information

David Fisher
Affiliation: Department of Mathematics, Rawles Hall, Indiana University, Bloomington, Indiana 47405

DOI: https://doi.org/10.1090/S0002-9947-07-04372-3
Received by editor(s): July 5, 2006
Published electronically: July 20, 2007
Additional Notes: The author was partially supported by NSF grant DMS-0226121 and a PSC-CUNY grant.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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