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


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.


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 



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.


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Copyright information

© © European Union 2014

Authors and Affiliations

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

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