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A geodesic lamination L on a closed hyperbolic surface S, when provided with a transverse measure μ, gives rise to a “dual R-tree” Tμ, together with an action of G = π1S on Tμ by isometries. A point of Tμ corresponds precisely to a leaf of the lift L̃ of L to the universal covering S̃ of S, or to a complementary component of L̃ in S̃. The G-action on T is… (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 L̃ of L to the universal covering of S) an action of π1S on an R-tree T which is often called dual to the lamination L̃. Conversely, every small action of π1S on an R-tree T comes from this construction, provided the surface is closed and the action… (More)

Let FN be a free group of rank N ≥ 2, let μ be a geodesic current on FN and let T be an R-tree with a very small isometric action of FN . 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(T ) of T . Applying this result, we obtain a generalization of a… (More)

- Ilya Kapovich, Martin Lustig
- J. London Math. Society
- 2015

For any atoroidal iwip φ ∈ Out(FN ), the mapping torus group Gφ = FN φ 〈t〉 is hyperbolic, and, by a result of Mitra, the embedding ι : FN −→ Gφ induces a continuous, FN -equivariant and surjective Cannon–Thurston map ι̂ : ∂FN → ∂Gφ. We prove that for any φ as above, the map ι̂ is finite-to-one and that the preimage of every point of ∂Gφ has cardinality at… (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 Heegaard surface Σ, which have at most two Dehn twists which fail to yield a hyperbolic manifold, is generic in… (More)

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 topology (or, equivalently, in the length… (More)

- Martin Lustig, Yoav Moriah
- 2002

In this paper we show that for a given 3-manifold and a given Heegaard splitting there are finitely many preferred decomposing systems of 3g − 3 disjoint essential disks. These are characterized by a combinatorial 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)

A concrete family of automorphisms αn of the free group Fn is exhibited, for any n ≥ 3, and the following properties are proved: αn is irreducible with irreducible powers, has trivial fixed subgroup, and has 2n− 1 attractive as well as 2n repelling fixed points at ∂Fn . As a consequence of a recent result of V Guirardel there can not be more fixed points on… (More)

Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any “sufficiently large” subgroup of Out(FN ) (that is, a subgroup containing a hyperbolic iwip). As a corollary we prove that for N ≥ 3… (More)

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