Free group automorphisms with parabolic boundary orbits

For $N\geq 4$, we show that there exist automorphisms of the free group $F_N$ which have a parabolic orbit in $\partial F_N$. In fact, we exhibit a technology for producing infinitely many such examples

Growth and order of automorphisms of free groups and free Burnside groups

We prove that an outer automorphism of the free group is exponentially growing if and only if it induces an outer automorphism of infinite order of free Burnside groups with sufficiently large odd exponent.Comment: 36 pages, 4 figure

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

Geometric realizations of two dimensional substitutive tilings

We define 2-dimensional topological substitutions. A tiling of the Euclidean plane, or of the hyperbolic plane, is substitutive if the underlying 2-complex can be obtained by iteration of a 2-dimensional topological substitution. We prove that there is no primitive substitutive tiling of the hyperbolic plane $\mathbb{H}^2$. However, we give an example of substitutive tiling of \Hyp^2 which is non-primitive.Comment: 30 pages, 13 figure

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

On the Maximal Subgroup of Automorphisms of a Free Group FN which Fix a Point of the Boundary FN

International audienceWe prove that the stabilizer of an attractive fixed point in ∂F N of an irreducible with irreducible powers automorphism of a free group F N is virtually cyclic. The proof uses the attractive lamination of such an automorphism. We also establish the complete list of all possible maximal subgroups of Aut(F 2) which fix a point of the boundary ∂F 2

Lamination duale à un arbre réel

International audienceNous présentons des résultats reliant un arbre réel muni d'une action par isométries du groupe libre, sa lamination duale et les courants portés par cette dernière

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

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow \Gamma$ and vector towers $\overleftarrow v$ on $\overleftarrow \Gamma$, which allows us to efficiently describe all invariant measures $\mu = \mu^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $\varphi: F_N \to F_N$ is determined, when $\varphi$ is represented by a train track map.Comment: 52 pages, 3 figures. This is rather a new paper than a new version of the old one. The setting is much more general, and also closer to a symbolic dynamics spirit. Also, some of the work from the original paper has been removed and will be taken up in a forthcoming paper. Accepted in Journal of London Mathematical society. To appear 201