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Article Dans Une Revue Groups, Geometry, and Dynamics Année : 2014

Rips Induction: Index of the dual lamination of an $\R$-tree

Thierry Coulbois
Arnaud Hilion

Résumé

Let $T$ be a $\R$-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 $\R$-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.
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Dates et versions

hal-01481865 , version 1 (03-03-2017)

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Thierry Coulbois, Arnaud Hilion. Rips Induction: Index of the dual lamination of an $\R$-tree. Groups, Geometry, and Dynamics, 2014, 8, pp.97 - 134. ⟨10.4171/GGD/218⟩. ⟨hal-01481865⟩
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