On free actions on R-trees

Group actions on R-trees arise from the study of hyperbolic geometry, and it is a natural generalization of Bass-Serre theory. The first case to consider is that of free actions. We are interested in two problems. What is the maximum number of orbits of branch points of free actions on R-trees? Which groups admit free actions on R-trees? The finiteness of branch points of free actions on R-trees is proved for a family of groups. This family contains free products of free abelian groups and surface groups, and it is closed under taking free product with amalgamation. Concerning the second problem, freely branched actions are defined. It is proved that the only groups admit this kind of actions are free products of free abelian groups. The values of translation length functions and splitting of groups are also discussed.