Locally compact convergence groups and n-transitive actions

TL;DR

This paper classifies certain locally compact groups acting on compact spaces, showing the non-existence of 2-transitive actions, characterizing 3-transitive actions as hyperbolic boundary actions, and describing properties for higher n.

Contribution

It extends Bowditch's hyperbolic group characterization to locally compact groups and clarifies the structure of n-transitive actions for various n.

Findings

No 2-transitive actions exist for infinite disconnected spaces.

3-transitive actions correspond to hyperbolic groups acting on their boundaries.

For n>3, the space has a local cut point under certain group 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.