Rips induction: index of the dual lamination of an R\mathbb{R}-tree

  • Thierry Coulbois

    Aix-Marseille Université, Marseille, France
  • Arnaud Hilion

    Aix-Marseille Université, Marseille, France


Let TT be a R\mathbb{R}-tree in the boundary of the Outer Space CVN_N, with dense orbits. The Q\mathcal{Q}-index of TT is defined by means of the dual lamination of TT. It is a generalisation of the Poincaré Lefschetz index of a foliation on a surface. We prove that the Q\mathcal{Q}-index of TT is bounded above by 2N22N-2, and we study the case of equality. The main tool is to develop the Rips machine in order to deal with systems of isometries on compact R\mathbb{R}-trees.

Combining our results on the Q\mathcal{Q}-index with results on the classical geometric index of a tree, developed by Gaboriau and Levitt, we obtain a beginning classification of trees.

Cite this article

Thierry Coulbois, Arnaud Hilion, Rips induction: index of the dual lamination of an R\mathbb{R}-tree. Groups Geom. Dyn. 8 (2014), no. 1, pp. 97–134

DOI 10.4171/GGD/218