Index realization for automorphisms of free groups

For any surface $\Sigma$ of genus $g \geq 1$ and (essentially) any collection of positive integers $i_1, i_2, \ldots, i_\ell$ with $i_1+\cdots +i_\ell = 4g-4$ Masur and Smillie have shown that there exists a pseudo-Anosov homeomorphism $h:\Sigma \to \Sigma$ with precisely $\ell$ singularities $S_1, \ldots, S_\ell$ in its stable foliation $\cal L$, such that $\cal L$ has precisely $i_k+2$ separatrices raying out from each $S_k$. In this paper we prove the analogue of this result for automorphisms of a free group $F_N$, where "pseudo-Anosov homeomorphism" is replaced by "fully irreducible automorphism" and the Gauss-Bonnet equality $i_1+\cdots +i_\ell = 4g-4$ is replaced by the index inequality $i_1+\cdots +i_\ell \leq 2N-2$ from Gaboriau, Jaeger, Levitt and Lustig.Comment: 19 pages, 3 figures, 1 table, revised version for publicatio

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

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

Botany of irreducible automorphisms of free groups

We give a classification of iwip outer automorphisms of the free group, by discussing the properties of their attracting and repelling trees.Comment: 13 pages. This paper was originally part of arXiv:1002.0972. Minor corrections from v2, numberings are consistant with published version (and title upgrade

R\R-trees and laminations for free groups II: The dual lamination of an R\R-tree

This is the second part of a series of three articles which introduce laminations for free groups (see math.GR/0609416 for the first part). Several definition of the dual lamination of a very small action of a free group on an $\R$-tree are given and proved to be equivalent.Comment: corrections of typos and minor updat

Non-unique ergodicity, observers' topology and the dual algebraic lamination for R\R-trees

We continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology.Comment: to appear in the Illinois Journal of Mat

Fractal trees for irreducible automorphisms of free groups

The self-similar structure of the attracting subshift of a primitive substitution is carried over to the limit set of the repelling tree in the boundary of Outer Space of the corresponding irreducible outer automorphism of a free group. Thus, this repelling tree is self-similar (in the sense of graph directed constructions). Its Hausdorff dimension is computed. This reveals the fractal nature of the attracting tree in the boundary of Outer Space of an irreducible outer automorphism of a free group

R\R-trees, dual laminations, and compact systems of partial isometries

Let \FN be a free group of finite rank $N \geq 2$, and let $T$ be an $\R$-tree with a very small, minimal action of \FN with dense orbits. For any basis \CA of \FN there exists a {\em heart} K_{\CA} \subset \bar T (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}, for each a_{i} \in \CA, defines a tree T_{(K_{\CA}, \CA)} which contains an isometric copy of $T$ as minimal subtree.Comment: minor updat

Non-unique ergodicity, observers' topology and the dual algebraic lamination for R\R-trees

International audienceWe continue in this article the study of laminations dual to very small actions of a free group F on R-trees. We prove that this lamination determines completely the combinatorial structure of the R-tree (the so-called observers' topology). On the contrary the metric is not determined by the lamination, and an R-tree may be equipped with different metrics which have the same observers' topology