THE RANK OF ACTIONS ON R-TREES

For n greater than or equal to 2, let F-n denote the free group of rank n. We define a total branching index i for a minimal small action of F-n on an R-tree. We show i less than or equal to 2n - 2, with equality if and only if the action is geometric. We thus recover Jiang's bound 2n - 2 for the number of orbits of branch points of free F-n-actions, and we extend it to very small actions (i.e, actions which are limits of free actions). The Q-rank of a minimal very small action of F, is bounded by 3n - 3, equality being possible only if the action is free simplicial. There exists a free action of F-3 such that the values of the length function do not lie in any finitely generated subgroup of R. The boundary of Culler-Vogtmann's outer space Y-n has topological dimension 3n - 5.