Advertisement

Geometriae Dedicata

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

Rigid geometric structures, isometric actions, and algebraic quotients

  • Jinpeng An
Original Paper

Abstract

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.

Keywords

Rigid geometric structure Isometric action Invariant measure Fundamental group Algebraic quotient 

Mathematics Subject Classification (2000)

53C15 57S20 37B05 37C85 20G20 22D40 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Adams S.: Dynamics on Lorentz Manifolds. World Scientific Publishing Co. Inc., River Edge, NJ (2001)zbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Borel A.: Density properties for certain subgroups of semi-simple groups without compact components. Ann. Math. 72(2), 179–188 (1960)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Borel A.: Linear Algebraic Groups, 2nd edn. Springer, New York (1991)zbMATHCrossRefGoogle Scholar
  9. 9.
    Candel A., Quiroga-Barranco R.: Gromov’s centralizer theorem. Geom. Dedicata 100, 123–155 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Candel A., Quiroga-Barranco R.: Parallelisms, prolongations of Lie algebras and rigid geometric structures. Manuscripta Math. 114(3), 335–350 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    D’Ambra G.: Isometry groups of Lorentz manifolds. Invent. Math. 92(3), 555–565 (1988)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981)MathSciNetzbMATHCrossRefGoogle 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)zbMATHGoogle 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)zbMATHCrossRefGoogle Scholar
  27. 27.
    Morris D.W.: Ratner’s Theorems on Unipotent Flows. University of Chicago Press, Chicago, IL (2005)zbMATHGoogle Scholar
  28. 28.
    Nevo A., Zimmer R.J.: Invariant rigid geometric structures and smooth projective factors. Geom. Funct. Anal. 19(2), 520–535 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Platonov V., Rapinchuk A.: Algebraic Groups and Number Theory. Academic Press Inc., Boston, MA (1994)zbMATHGoogle Scholar
  30. 30.
    Ratner M.: Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math. 101(2), 449–482 (1990)MathSciNetzbMATHCrossRefGoogle 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)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Stuck G.: Minimal actions of semisimple groups. Ergodic Theory Dynam. Syst. 16(4), 821–831 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Varadarajan V.S.: Groups of automorphisms of Borel spaces. Trans. Am. Math. Soc. 109, 191–220 (1963)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Varadarajan V.S.: Lie Groups, Lie Algebras, and Their Representations. Springer, New York (1984)zbMATHGoogle Scholar
  36. 36.
    Zeghib A.: On Gromov’s theory of rigid transformation groups: a dual approach. Ergodic Theory Dynam. Syst. 20, 935–946 (2000)MathSciNetzbMATHCrossRefGoogle 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)zbMATHGoogle 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)MathSciNetzbMATHGoogle 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)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina

Personalised recommendations