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

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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.

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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.

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Authors and Affiliations

  1. Université catholique de Louvain, IRMP, Chemin du Cyclotron 2, bte L7.01.01, 1348 , Louvain-la-Neuve, Belgium

    Mathieu Carette

  2. School of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, UK

    Dennis Dreesen

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  1. Mathieu Carette
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  2. Dennis Dreesen
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Correspondence to Dennis Dreesen.

Additional information

The first author is a Postdoctoral Researcher of the F.R.S.-FNRS (Belgium).

The major part of this work was conducted while the second author was staying at the Université cathologique de Louvain in Louvain-la-Neuve. He gratefully acknowledges the support of the F.R.S.-FNRS (Belgium), Grant F.4520.11. Currently, the second author is a Marie Curie Intra-European Fellow within the 7th European Community Framework Programme.

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Carette, M., Dreesen, D. Locally compact convergence groups and \(n\)-transitive actions. Math. Z. 278, 795–827 (2014). https://doi.org/10.1007/s00209-014-1334-2

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