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- François Gautero, Martin Lustig
- 2004

We show that the semi-direct product of a one-ended torsion-free word-hyperbolic group with Z, given through an automorphism α : G → G, is a hyperbolic group relative to certain canonical subgroups of G on which α acts periodically or with linear growth.

For the free group FN of finite rank N ≥ 2 we construct a canonical Bonahon-type, continuous and Out(FN)-invariant geometric intersection form , : cv(FN) × Curr(FN) → R ≥0. Here cv(FN) is the closure of unprojectivized Culler-Vogtmann's Outer space cv(FN) in the equivariant Gromov-Hausdorff convergence topol-ogy (or, equivalently, in the length function… (More)

Let F N be a free group of rank N ≥ 2, let µ be a geodesic current on F N and let T be an 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… (More)

- MARTIN LUSTIG, YOAV MORIAH
- 2009

We introduce a notion of " genericity " for countable sets of curves in the curve complex of a surface Σ, based on the Lebesgue measure on the space of projective measured laminations in Σ. With this definition we prove that the set of curves on a Hee-gaard surface Σ, which have at most two Dehn twists which fail to yield a hyperbolic manifold, is generic… (More)

- Thierry Coulbois, Arnaud Hilion, Martin Lustig
- 2008

A geodesic lamination L on a hyperbolic surface S, provided with a transverse measure, defines (via the lift L of L to the universal covering of S) an action of π 1 S on an R-tree T which is often called dual to the lamination L. Conversely, every small action of π 1 S on an R-tree T comes from this construction, provided the surface is closed and the… (More)

A geodesic lamination L on a closed hyperbolic surface S, when provided with a transverse measure µ, gives rise to a " dual R-tree " T µ , together with an action of G = π 1 S on T µ by isometries. A point of T µ corresponds precisely to a leaf of the lift L of L to the universal covering S of S, or to a complementary component of L in S. The G-action on T… (More)

We show that every automorphism α of a free group F k of finite rank k has asymptotically periodic dynamics on F k and its boundary ∂F k : there exists a positive power α q such that every element of the compactum F k ∪ ∂F k converges to a fixed point under iteration of α q. Further results about the dynamics of α as well as an extension from F k to… (More)

Let T be an R-tree with a very small action of a free group F N which has dense orbits. Such a tree T or its metric completion T are not locally compact. However, if one adds the Gromov boundary ∂T to T , then there is a coarser observers' topology on the union T ∪ ∂T , and it is shown here that this union, provided with the observers' topology, is a… (More)

- Martin Lustig, Yoav Moriah
- 2002

In this paper we show that for a given 3-manifold and a given Hee-gaard splitting there are finitely many preferred decomposing systems of 3g − 3 disjoint essential disks. These are characterized by a combi-natorial criterion which is a slight strengthening of Casson-Gordon's rectangle condition. This is in contrast to fact that in general there can exist… (More)

- Martin Lustig
- 2009

Let FN be a free group of finite rank N ≥ 2, and let T be an R-tree with a very small, minimal action of FN with dense orbits. For any basis A of FN there exists a heart KA ⊂ T (= the metric completion of T) which is a compact subtree that has the property that the dynamical system of partial isometries ai : KA ∩aiKA → a −1 i KA ∩KA, for each ai ∈ A,… (More)