R-Trees and Normalization of Pseudogroups

Let be a pseudogroup defined on a tree Z, and let Λ be a finite set of generators for . The reduced fundamental group (Λ) of Λ is defined here. I give a new and experimentally inspired proof of a result of Levitt : If (Λ) is a free group, there exists a finite set of generators Ψ for such that is free on the set Ψ. If Ψ has no dead ends, it is an interval exchange. Like Gaboriau, Levitt and Paulin [Gaboriau et al, 19921. I prove that if G is a finitely presented group acting freely on an R tree and Λ is a corresponding set of pseudogroup generators, we're in one of the following situations: either G splits as a free product with a noncyclic free abelian summand, or Λ can be reduced to an interval exchange by normalizing and removing a finite number of dead ends, or the process of removing dead ends from Λ does not terminate in a finite number of steps.