R-Trees and Normalization of Pseudogroups

Frank Rimlinger

Abstract

Let 𝒢 be a pseudogroup defined on a tree Z, and let Γ be a finite set of generators for 𝒢. The reduced fundamental group π̄(Γ) of Γ is defined here. I give a new and experimentally inspired proof of a result of Levitt: If π̄(Γ) is a free group, there exists a finite set of generators Ψ for 𝒢 such that π̄(Ψ) 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. 1992], I prove that if G is a finitely presented group acting freely on an ℝ-tree and Γ 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 Γ 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 Γ does not terminate in a finite number of steps.