Geometriae Dedicata

, Volume 157, Issue 1, pp 153–185 | Cite as

Rigid geometric structures, isometric actions, and algebraic quotients

Original Paper


By using a Borel density theorem for algebraic quotients, we prove a theorem concerning isometric actions of a Lie group G on a smooth or analytic manifold M with a rigid A-structure σ. It generalizes Gromov’s centralizer and representation theorems to the case where R(G) is split solvable and G/R(G) has no compact factors, strengthens a special case of Gromov’s open dense orbit theorem, and implies that for smooth M and simple G, if Gromov’s representation theorem does not hold, then the local Killing fields on \({\widetilde{M}}\) are highly non-extendable. As applications of the generalized centralizer and representation theorems, we prove (1) a structural property of Iso(M) for simply connected compact analytic M with unimodular σ, (2) three results illustrating the phenomenon that if G is split solvable and large then π 1(M) is also large, and (3) two fixed point theorems for split solvable G and compact analytic M with non-unimodular σ.


Rigid geometric structure Isometric action Invariant measure Fundamental group Algebraic quotient 

Mathematics Subject Classification (2000)

53C15 57S20 37B05 37C85 20G20 22D40 


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  1. 1.
    Adams S.: Dynamics on Lorentz Manifolds. World Scientific Publishing Co. Inc., River Edge, NJ (2001)MATHCrossRefGoogle Scholar
  2. 2.
    Bader U., Frances C., Melnick K.: An embedding theorem for automorphism groups of Cartan geometries. Geom. Funct. Anal. 19(2), 333–355 (2009)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Benoist Y.: Orbites des structures rigides (d’aprs M. Gromov), Integrable systems and foliations, pp. 1–17. Birkhäuser, Boston (1997)CrossRefGoogle Scholar
  4. 4.
    Benoist Y., Foulon P., Labourie F.: Flots d’Anosov à distributions stable et instable différentiables. J. Am. Math. Soc. 5(1), 33–74 (1992)MathSciNetMATHGoogle Scholar
  5. 5.
    Benoist Y., Labourie F.: Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables. Invent. Math. 111(2), 285–308 (1993)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Benveniste E.J., Fisher D.: Nonexistence of invariant rigid structures and invariant almost rigid structures. Comm. Anal. Geom. 13(1), 89–111 (2005)MathSciNetMATHGoogle Scholar
  7. 7.
    Borel A.: Density properties for certain subgroups of semi-simple groups without compact components. Ann. Math. 72(2), 179–188 (1960)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Borel A.: Linear Algebraic Groups, 2nd edn. Springer, New York (1991)MATHCrossRefGoogle Scholar
  9. 9.
    Candel A., Quiroga-Barranco R.: Gromov’s centralizer theorem. Geom. Dedicata 100, 123–155 (2003)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Candel A., Quiroga-Barranco R.: Parallelisms, prolongations of Lie algebras and rigid geometric structures. Manuscripta Math. 114(3), 335–350 (2004)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    D’Ambra G.: Isometry groups of Lorentz manifolds. Invent. Math. 92(3), 555–565 (1988)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    D’Ambra G., Gromov M.: Lectures on transformation groups: geometry and dynamics, Surveys in differential geometry, pp. 19–111. Lehigh University, Bethlehem, PA (1991)Google Scholar
  13. 13.
    Dani S.G.: On ergodic quasi-invariant measures of group automorphism. Israel J. Math. 43, 62–74 (1982)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dumitrescu S.: Meromorphic almost rigid geometric structures. Geometry, Rigidity, and Group Actions. University of Chicago Press, Chicago, IL (2011)Google Scholar
  15. 15.
    Effros E.G.: Transformation groups and C*-algebras. Ann. Math. 81(2), 38–55 (1965)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Feres, R.: Rigid geometric structures and actions of semisimple Lie groups, Rigidité, groupe fondamental et dynamique, pp. 121–167, Soc. Math. France, Paris (2002)Google Scholar
  17. 17.
    Feres, R., Katok, A.: Ergodic theory and dynamics of G-spaces (with special emphasis on rigidity phenomena). Handbook of dynamical systems, vol. 1A, pp. 665–763, North-Holland, Amsterdam (2002)Google Scholar
  18. 18.
    Fisher D.: Groups acting on manifolds: around the Zimmer program Geometry, Rigidity, and Group Actions. University of Chicago Press, Chicago, IL (2011)Google Scholar
  19. 19.
    Fisher D., Zimmer R.J.: Geometric lattice actions, entropy and fundamental groups. Comment. Math. Helv. 77(2), 326–338 (2002)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Gromov M.: Rigid Transformations Groups, Géométrie Différentielle, pp. 65–139. Hermann, Paris (1988)Google Scholar
  22. 22.
    Knapp A.W.: Lie Groups Beyond an Introduction, 2nd edn. Birkhäuser Boston Inc., Boston, MA (2002)MATHGoogle Scholar
  23. 23.
    Kobayashi S., Nomizu K.: Foundations of Differential Geometry, vol. I. Wiley, New York (1996)Google Scholar
  24. 24.
    Labourie, F.: Large groups actions on manifolds. In: Proceedings of the International Congress of Mathematicians, vol. II (Berlin, 1998), Doc. Math., 1998, Extra vol. II, pp. 371–380Google Scholar
  25. 25.
    Melnick, K.: A Frobenius theorem for Cartan geometries, with applications (preprint)Google Scholar
  26. 26.
    Moerdijk I., Mrčun J.: Introduction to Foliations and Lie Groupoids. Cambridge University Press, Cambridge (2003)MATHCrossRefGoogle Scholar
  27. 27.
    Morris D.W.: Ratner’s Theorems on Unipotent Flows. University of Chicago Press, Chicago, IL (2005)MATHGoogle Scholar
  28. 28.
    Nevo A., Zimmer R.J.: Invariant rigid geometric structures and smooth projective factors. Geom. Funct. Anal. 19(2), 520–535 (2009)MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Platonov V., Rapinchuk A.: Algebraic Groups and Number Theory. Academic Press Inc., Boston, MA (1994)MATHGoogle Scholar
  30. 30.
    Ratner M.: Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math. 101(2), 449–482 (1990)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Rosenlicht M.: A remark on quotient spaces. An. Acad. Brasil. Ci. 35, 487–489 (1963)MathSciNetGoogle Scholar
  32. 32.
    Shalom Y.: Invariant measures for algebraic actions, Zariski dense subgroups and Kazhdan’s property (T). Trans. Am. Math. Soc. 351, 3387–3412 (1999)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Stuck G.: Minimal actions of semisimple groups. Ergodic Theory Dynam. Syst. 16(4), 821–831 (1996)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Varadarajan V.S.: Groups of automorphisms of Borel spaces. Trans. Am. Math. Soc. 109, 191–220 (1963)MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Varadarajan V.S.: Lie Groups, Lie Algebras, and Their Representations. Springer, New York (1984)MATHGoogle Scholar
  36. 36.
    Zeghib A.: On Gromov’s theory of rigid transformation groups: a dual approach. Ergodic Theory Dynam. Syst. 20, 935–946 (2000)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Zeghib, A.: Sur les groupes de transformations rigides: théorème de l’orbite dense-ouverte, Rigidité, groupe fondamental et dynamique, 169–188, Soc. Math. France, Paris (2002)Google Scholar
  38. 38.
    Zimmer R.J.: Ergodic Theory and Semisimple Groups. Birkhäuser Verlag, Basel (1984)MATHGoogle Scholar
  39. 39.
    Zimmer, R.J.: Actions of semisimple groups and discrete subgroups, Proceedings of the International Congress of Mathematicians, (Berkeley, Calif., 1986), pp. 1247–1258, Am. Math. Soc., Providence, RI (1987)Google Scholar
  40. 40.
    Zimmer R.J.: Split rank and semisimple automorphism groups of G-structures. J. Differ. Geom. 26(1), 169–173 (1987)MathSciNetMATHGoogle Scholar
  41. 41.
    Zimmer, R.J.: Automorphism groups and fundamental groups of geometric manifolds. In: Differential geometry: Riemannian geometry, 693–710, Proceedings of Sympososium on Pure Mathematics, 54, Part 3, Am. Math. Soc., Providence, RI (1993)Google Scholar
  42. 42.
    Zimmer R.J., Morris D.W.: Ergodic Theory, Groups, and Geometry. American Mathematical Society, Providence, RI (2008)MATHGoogle Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina

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