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

TL;DR

This paper introduces the $Q$-index for $ ext{R}$-trees in the boundary of Outer Space, proves an upper bound, and uses the Rips Machine to classify trees and automorphisms of free groups.

Contribution

It develops the Rips Machine for systems of isometries on compact $ ext{R}$-trees and establishes bounds on the $Q$-index, advancing the classification of $ ext{R}$-trees and automorphisms.

Findings

The $Q$-index of an $ ext{R}$-tree is bounded above by $2N-2$.

Developed the Rips Machine for systems of isometries on compact $ ext{R}$-trees.

Provided a classification framework for iwip outer automorphisms of free groups.

Abstract

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.