We show how particular kinds of group actions on topological trees give rise to isometric actions on $ {\Bbb R} $-trees. As a consequence, one can deduce results about group splittings. For example, we show that if $ \Gamma $ is a finitely presented infinite group such that any ascending chain of finite subgroups eventually stabilises and $ \Gamma $ admits a convergence action on a dendron, then $ \Gamma $ splits over a finite or two-ended subgroup. Using this result one can show that if $ \Gamma $ is a one-ended hyperbolic group with a global cut point in its boundary, then $ \Gamma $ splits over a two-ended subgroup. This is a step towards proving that no such cut point can exist. The paper gives an account of the relation between group actions on trees and foliations of 2-complexes.
Topology 37 (1998) 1275-1298.