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 $Q$-index of $T$ is defined by means of the dual lamination of $T$. It is a generalisation of the Euler-Poincaré index of a foliation on a surface. We prove that the $Q$-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 $\CQ$-index with results on the classical geometric index of a tree, we obtain a beginning of classification of trees. As a consequence, we give a classification of iwip outer automorphisms of the free group, by discussing the properties of their attracting and repelling trees.

arXiv comment33 pages. The previous version has been splitted in two disjoint papers. See also Botanic of irreducible automorphisms of free groups

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