# Groups, Geometry, and Dynamics

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**Volume 8, Issue 1, 2014, pp. 97–134**

**DOI: 10.4171/GGD/218**

Published online: 2014-05-13

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

Thierry Coulbois^{[1]}and Arnaud Hilion

^{[2]}(1) Aix-Marseille Université, Marseille, France

(2) Aix-Marseille Université, Marseille, France

Let $T$ be a $\mathbb{R}$-tree in the boundary of the Outer Space CV$_N$, with dense orbits. The $\mathcal{Q}$-index of $T$ is defined by means of the dual lamination of $T$. It is a generalisation of the Poincaré Lefschetz index of a foliation on a surface. We prove that the $\mathcal{Q}$-index of $T$ is bounded above by $2N-2$, 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 $\mathbb{R}$-trees.

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

*Keywords: *$\mathbb{R}$-trees, Outer Space, Rips machine

Coulbois Thierry, Hilion Arnaud: Rips induction: index of the dual lamination of an $\mathbb{R}$-tree. *Groups Geom. Dyn.* 8 (2014), 97-134. doi: 10.4171/GGD/218