Locally compact convergence groups and n-transitive actions
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 give a characterization of non-compact groups acting 3-properly and transitively on infinite compact sets as non-elementary boundary transitive hyperbolic groups. The main tool is a generalization to locally compact groups of Bowditch's topological characterization of hyperbolic groups. Finally, in contrast to the case n=3, we show that for n>3, if a locally compact group acts continuously, n-properly and n-cocompactly on a locally connected metrizable compactum M, then M has a local cut point.
Cite
@article{arxiv.1212.4021,
title = {Locally compact convergence groups and n-transitive actions},
author = {Mathieu Carette and Dennis Dreesen},
journal= {arXiv preprint arXiv:1212.4021},
year = {2014}
}