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On dense embeddings of discrete groups into locally compact groups

Abstract

We consider a class of discrete groups which have no ergodic actions by translations on continuous non-compact locally compact groups. We also study dense embeddings ofZ n (n>1) into non-compact locally compact groups. Moreover, we study some discrete groups which admit no embeddings into almost connected locally compact groups. In particular, we prove that a lattice in a simple Lie group with property (T) cannot be embedded densely into a connected non-compact locally compact group.

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References

  1. 1.

    N. Bourbaki,Groupes et Algébres de Lie, Ch. I–III, Hermann, Paris, 1971.

  2. 2.

    J. Cleary andS.A. Morris, Generators for locally compact groups,Proc. of the Edinburgh Math. Soc.,10 (1993), 463–467.

  3. 3.

    K. Corlette, Archimedean superrigidity and hyperboliques geometry,Ann. of Math.,135 (1992), 165–182.

  4. 4.

    S.L. Gefter andV.Ya. Golodets, Fundamental groups for ergodic actions and actions with unit fundamental groups,Publ. RIMS, Kyoto Univ.,24 (1988), 821–847.

  5. 5.

    S.L. Gefter andK.M. Kulagin, On dense embeddings of discrete Abelian groups into locally compact groups,Bull. Belg. Math. Soc.,9 (2002), 161–165.

  6. 6.

    E. Hewitt andK.A. Ross,Abstract Harmonic analysis, vol. 1, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.

  7. 7.

    I. Kaplansky,An introduction to differential algebra, Herman, Paris, 1957.

  8. 8.

    A. Lubotzky,Discrete groups, expanding graphs and invariant measures, Verlag, Basel-Boston-Berlin, 1994.

  9. 9.

    A. Lubotzky andB. Weiss, Groups and expanders,DIMACS Ser. in Discrete Math. and Th. Computer Science, vol. 10, 1993, 95–109.

  10. 10.

    G.W. Mackey, Ergodic transformation groups with a pure point spectrum,Ill. J. Math.,8 (1964), 593–600.

  11. 11.

    G.A. Margulis,Discrete subgroups of semisimple Lie groups, Springer, Berlin, 1991.

  12. 12.

    Ju.I. Merzljakov,Rational groups, (in Russian), Nauka, Moscow, 1980.

  13. 13.

    S.A. Morris,Pontryagin duality and the structure of locally compact Abelian groups, Cambridge University Press, Cambridge, 1977.

  14. 14.

    D. Montgomery andL. Zippin,Topological transformation groups, Interscience, New York-London, 1965.

  15. 15.

    J-P. Serre,Trees, Springer-Verlag, New York, 1980.

  16. 16.

    R.J. Zimmer, Normal ergodic actions,J. Funct. Anal.,25 (1977), 286–305.

  17. 17.

    R.J. Zimmer, Amenable actions and dense subgroups of Lie groups,J. Funct. Anal.,72 (1987), 58–64.

  18. 18.

    R.J. Zimmer,Ergodic theory and semisimple groups, Birkhäuser, Boston, 1985.

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Correspondence to Maxim S. Boyko.

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Boyko, M.S., Gefter, S.L. & Kulagin, K.M. On dense embeddings of discrete groups into locally compact groups. Qual. Th. Dyn. Syst. 4, 31 (2003). https://doi.org/10.1007/BF02972820

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Key words

  • locally compact groups
  • discrete groups
  • dense subgroups