Advertisement

Mathematische Zeitschrift

, Volume 278, Issue 3–4, pp 795–827 | Cite as

Locally compact convergence groups and \(n\)-transitive actions

  • Mathieu Carette
  • Dennis DreesenEmail author
Article

Abstract

All \(\sigma \)-compact, locally compact groups acting sharply \(n\)-transitively and continuously on compact spaces \(M\) have been classified, except for \(n=2,3\) when \(M\) is infinite and disconnected. We show that no such actions exist for \(n=2\) and that these actions for \(n=3\) coincide with the action of a hyperbolic group on a space equivariantly homeomorphic to its hyperbolic boundary. We further characterize non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary-transitive hyperbolic groups, which in turn were recently studied by Caprace, de Cornulier, Monod and Tessera. As an important tool, we generalize Bowditch’s topological characterization of discrete hyperbolic groups to locally compact hyperbolic groups. Finally, we show that if a locally compact group acts continuously, 4-properly and 4-cocompactly on a locally connected metrizable compactum M, then M has a global cut point, which is in sharp contrast to the \(3\)-proper, \(3\)-cocompact case due to the solution of Bowditch’s cut-point conjecture.

Keywords

Hyperbolic groups Geometric group theory Locally compact groups Multiply transitive actions 

Mathematics Subject Classification (2010)

Primary 20F67 (Hyperbolic groups and non-positively curved groups ) Secondary 20F69  20B22 22D05 

Notes

Acknowledgments

The authors are grateful to Pierre-Emmanuel Caprace for suggesting the problem, as well as for helpful discussions. The authors also thank him and Ralf Köhl for motivating discussions which lead to Theorem D. We also thank a referee for many interesting comments.

References

  1. 1.
    Arens, R.: Topologies for homeomorphism groups. Am. J. Math. 68, 593–610 (1946)Google Scholar
  2. 2.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)Google Scholar
  3. 3.
    Bass, H., Kulkarni, R.: Uniform tree lattices. J. Am. Math. Soc. 3(4), 843–902 (1990)Google Scholar
  4. 4.
    Bass, H., Lubotzky, A.: Tree Lattices, Progress in Mathematics, vol. 176, Birkhäuser Boston Inc, Boston (2001). With appendices by Bass, L. Carbone, Lubotzky, G. Rosenberg and J. TitsGoogle Scholar
  5. 5.
    Bestvina, M., Mess, G.: The boundary of negatively curved groups. J. Am. Math. Soc. 4(3), 469–481 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Bourbaki, N.: Éléments de mathématique. Fascicule XXIX. Livre VI: Intégration. Chapitre 7: mesure de Haar. Chapitre 8: convolution et représentations, Actualités Scientifiques et Industrielles, No. 1306, Hermann, Paris (1963)Google Scholar
  7. 7.
    Bowditch, B.H.: A topological characterisation of hyperbolic groups. J. Am. Math. Soc. 11(3), 643–667 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Bowditch, B.H.: Connectedness properties of limit sets. Trans. Am. Math. Soc. 351(9), 3673–3686 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Bowditch, B.H.: Convergence Groups and Configuration Spaces, Geometric Group Theory Down Under (Canberra: de Gruyter 1996), pp. 23–54. Berlin (1999)Google Scholar
  10. 10.
    Caprace, P.-E., de Cornulier, Y., Monod, N., Tessera, R.: Amenable hyperbolic groups. J. Eur. Math. Soc. (to appear)Google Scholar
  11. 11.
    Caprace, P.-E., De Medts, T.: Trees, contraction groups, and Moufang sets. Duke Math. J. 162(13), 2413–2449 (2013)Google Scholar
  12. 12.
    Casson, A., Jungreis, D.: Convergence groups and Seifert fibered \(3\)-manifolds. Invent. Math. 118(3), 441–456 (1994)Google Scholar
  13. 13.
    Cornulier, Y.: On the quasi-isometric classification of focal hyperbolic groups, (2012, preprint). http://arxiv.org/abs/1212.2229v1
  14. 14.
    Gabai, D.: Convergence groups are Fuchsian groups. Ann. Math. (2) 136(3), 447–510 (1992)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Ghys, É., de la Harpe, P. (eds.): Sur les Groupes Hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston Inc, Boston (1990). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, (1988)Google Scholar
  16. 16.
    Gerasimov, V.: Expansive convergence groups are relatively hyperbolic. Geom. Funct. Anal. 19(1), 137–169 (2009)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Gromov, M.: Hyperbolic Groups, Essays in Group Theory, Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar
  18. 18.
    Hinkkanen, A.: Abelian and nondiscrete convergence groups on the circle. Trans. Am. Math. Soc. 318(1), 87–121 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Jordan, C.: Recherches sur les substitutions. J. Math. Pures Appl. 17(2), 351–363 (1872)zbMATHGoogle Scholar
  20. 20.
    Kerby, W.: On Infinite Sharply Multiply Transitive Groups, Vandenhoeck and Ruprecht, Göttingen (1974). Hamburger Mathematische Einzelschriften. Neue Folge, Heft 6Google Scholar
  21. 21.
    Kakutani, S., Kodaira, K.: Über das Haarsche Mass in der lokal bikompakten Gruppe. Proc. Imp. Acad. Tokyo 20, 444–450 (1944)Google Scholar
  22. 22.
    Kramer, L.: Two-transitive Lie groups. J. Reine Angew. Math. 563, 83–113 (2003)Google Scholar
  23. 23.
    Mangesh, G.: Murdeshwar, General Topology, A Halsted Press Book. Wiley, New York (1983)Google Scholar
  24. 24.
    Nevo, A.: A structure theorem for boundary-transitive graphs with infinitely many ends. Isr. J. Math. 75(1), 1–19 (1991)Google Scholar
  25. 25.
    Serre, J.-P.: Trees. Springer, Berlin (1980). Translated from the French by John StillwellGoogle Scholar
  26. 26.
    Swarup, G.A.: On the cut point conjecture. Electron. Res. Announc. Am. Math. Soc. 2(2), 98–100 (electronic) (1996)Google Scholar
  27. 27.
    Tits, J.: Généralisations des groupes projectifs basées sur leurs propriétés de transitivité. Acad. R. Belgique. Cl. Sci. Mém. Coll. \(8^\circ \) 27(2), 115 (1952)Google Scholar
  28. 28.
    Tits, J.: Sur les groupes doublement transitifs continus. Comment. Math. Helv. 26, 203–224 (1952)CrossRefMathSciNetzbMATHGoogle Scholar
  29. 29.
    Tits, J.: Sur certaines classes d’espaces homogènes de groupes de Lie. Acad. R. Belg. Cl. Sci. Mém. Coll. \(8^\circ \) 29(3), 268 (1955)Google Scholar
  30. 30.
    Tits, J.: Sur le groupe des automorphismes d’un arbre, Essays on topology and related topics (Mémoires dédiés à Georges de Rham), pp. 188–211. Springer, New York (1970)Google Scholar
  31. 31.
    Tukia, P.: Homeomorphic conjugates of Fuchsian groups. J. Reine Angew. Math. 391, 1–54 (1988)Google Scholar
  32. 32.
    Tukia, P.: Conical limit points and uniform convergence groups. J. Reine Angew. Math. 501, 71–98 (1998)Google Scholar
  33. 33.
    Yaman, A.: Proper cocompact actions on configuration spaces (2003, preprint)Google Scholar
  34. 34.
    Zassenhaus, H.: Kennzeichnung endlicher linearer Gruppen als Permutationsgruppen. Abh. Math. Sem. Hambg. 11, 17–40 (1936)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Zassenhaus, H.: Über endliche Fastkörper. Abh. Math. Sem. Hambg 11, 187–220 (1936)CrossRefMathSciNetGoogle Scholar

Copyright information

© © European Union 2014

Authors and Affiliations

  1. 1.Université catholique de Louvain, IRMPLouvain-la-NeuveBelgium
  2. 2.School of MathematicsUniversity of SouthamptonSouthamptonUK

Personalised recommendations