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

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

2010-02-04

arxiv.org/abs/1002.0972v3

Mathematics

Group Theory

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

Scientific paper

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-Poincar\'e 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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