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

Abstract

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.