Thierry Coulbois, Arnaud Hilion, Martin Lustig
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.
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This is the third of a series of three articles where we introduce laminations for the free-groups. We explore here the link between currents and laminations and prove that the situation is more complicated than in the surface case of real tree dual to a measured geodesic lamination.
This paper is the first of a sequence of three papers, where the concept of an $\mathbb R$-tree dual to a measured geodesic lamination in a hyperbolic surface is generalized to arbitrary $\mathbb R$-trees provided with a (very small) action of the free group $F_N$ of finite rank $N\geq 2$ by isometries. Three different definitions are given and they are proved to be equivalent. We also describe the topology and Out$(F_N)$-action on the space of laminations.
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.
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.
For every atoroidal iwip automorphism $ϕ$ of $F_N$ (i.e. the analogue of a pseudo-Anosov mapping class) it is shown that the algebraic lamination dual to the forward limit tree $T_+(ϕ)$ is obtained as "diagonal closure" of the support of the backward limit current $μ_-(ϕ)$. This diagonal closure is obtained through a finite procedure in analogy to adding diagonal leaves from the complementary components to the stable lamination of a pseudo-Anosov homeomorphism. We also give several new characterizations as well as a structure theorem for the dual lamination of $T_+(ϕ)$, in terms of Bestvina-Fe…
This work is the first step towards a description of the Gromov boundary of the free factor graph of a free product, with applications to subgroup classification for outer automorphisms. We extend the theory of algebraic laminations dual to trees, as developed by Coulbois, Hilion, Lustig and Reynolds, to the context of free products; this also gives us an opportunity to give a unified account of this theory. We first show that any $\mathbb{R}$-tree with dense orbits in the boundary of the corresponding outer space can be reconstructed as a quotient of the boundary of the group by its dual lami…
We extend the techniques of [CH] to build an inductive procedure for studying actions in the boundary of the Culler-Vogtmann Outer Space, the main novelty being an adaptation of he classical Rauzy-Veech induction for studying actions of surface type. As an application, we prove that a tree in the boundary of Outer space is free and indecomposable if and only if its dual lamination is minimal up to diagonal leaves. Our main result generalizes [BFH97, Proposition 1.8] as well as the main result of [KL11].
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 …
Let $T$ be an $\mathbb{R}$-tree, equipped with a very small action of the rank $n$ free group $F_n$, and let $H \leq F_n$ be finitely generated. We consider the case where the action $F_n \curvearrowright T$ is indecomposable--this is a strong mixing property introduced by Guirardel. In this case, we show that the action of $H$ on its minimal invarinat subtree $T_H$ has dense orbits if and only if $H$ is finite index in $F_n$. There is an interesting application to dual algebraic laminations; we show that for $T$ free and indecomposable and for $H \leq F_n$ finitely generated, $H$ carries a le…
Let $F_N$ be a free group of rank $N\ge 2$, let $μ$ be a geodesic current on $F_N$ and let $T$ be an $\mathbb R$-tree with a very small isometric action of $F_N$. We prove that the geometric intersection number $<T, μ>$ is equal to zero if and only if the support of $μ$ is contained in the dual algebraic lamination $L^2(T)$ of $T$. Applying this result, we obtain a generalization of a theorem of Francaviglia regarding length spectrum compactness for currents with full support. As another application, we define the notion of a \emph{filling} element in $F_N$ and prove that filling elements are …
We prove that given a finite rank free group $\mathbb{F}$ of rank $\geq 3$ and two exponentially growing outer automorphisms $ψ$ and $ϕ$ with dual lamination pairs $Λ^\pm_ψ$ and $Λ^\pm_ϕ$ associated to them, and given a free factor system $\mathcal{F}$ with co-edge number $\geq 2$, $ϕ, ψ$ each preserving $\mathcal{F}$, so that the pair $(ϕ, Λ^\pm_ϕ), (ψ, Λ^\pm_ψ)$ is independent relative to $\mathcal{F}$, then there $\exists$ $M\geq 1$, such that for any integer $m,n \geq M$, the group $\langle ϕ^m, ψ^n \rangle$ is a free group of rank 2, all of whose non-trivial elements except perhaps the po…
When 1 -> H -> G -> Q -> 1 is a short exact sequence of three infinite, word-hyperbolic groups, Mahan Mitra (Mj) has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mitra associates to every point z in the Gromov boundary of Q an "ending lamination" on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a de…
Limit trees for free group automorphisms: universality
本文为自由群自同构关联了具有三个定义性质的普遍极限树，证明了该树上的作用能检测自同构的主导指数动态，并推论先前构造的极限预树具有典范性。
We study metric properties of maximal framed representations of fundamental groups of surfaces in symplectic groups over real closed fields, interpreted as actions on Bruhat-Tits buildings endowed with adapted Finsler norms. We prove that the translation length can be computed as intersection with a geodesic current, give sufficient conditions guaranteeing that such a current is a multicurve, and, if the current is a measured lamination, construct an isometric embedding of the associated tree in the building. These results are obtained as application of more general results of independent inte…
We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \leq \FN$ reduces $T$ if and only if $F$ acts on some subtree of $T$ with dense orbits. We characterize those trees, called arational, which do not admit a reduction by any free factor: $T$ is arational if and only if either $T$ is free and indecomposable or $T$ is dual to a surface with one boundary component equipped with an arational measured foliation. To complement this result, we establish so…
A spanning tree T in a finite planar connected graph G determines a dual spanning tree T* in the dual graph G such that T and T* do not intersect. We show that it is not always possible to find T in G, such that the diameters of T and T* are both within a uniform multiplicative constant (independent of G) of the diameters of their ambient graphs.
Let $T$ be an $\R$-tree in the boundary of Outer space with dense orbits. When the free group $\FN$ acts freely on $T$, we prove that the number of projective classes of ergodic currents dual to $T$ is bounded above by $3N-5$. We combine Rips induction and splitting induction to define unfolding induction for such an $\R$-tree $T$. Given a current $μ$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards $μ$. We also give a unique ergodicity criterion.
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