GRAPHS OF ACTIONS ON R-TREES

Let G be a finitely generated group acting on an R-tree T. First assum e that the action is free, and minimal (there is no proper invariant s ubtree), or more generally that it satisfies a certain finiteness cond ition. Then it may be described as a graph of transitive actions: the action may be recovered from a finite graph, together with additional data; in particular, every vertex v carries an action (G(v), T(v)) who se orbits are dense. For the action (G, T), it follows for instance th at the closure of any orbit is a discrete union of closed subtrees: it cannot meet a segment in a Cantor set. Now let l be the length functi on for an arbitrary action of G. For epsilon > 0 small enough, the sub group G(epsilon) subset-of G generated by elements g with l(g) less-th an-or-equal-to epsilon is independent of epsilon, and G/G(epsilon) is free. Several interpretations are given for the rank of G/G(epsilon).