Martin Lustig

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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)
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)
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)
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)
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)
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)