  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.
