Invariant Laminations for Irreducible Automorphisms of Free Groups

Kapovich, Ilya; Lustig, Martin

doi:10.1093/qmath/hat056

Cited by 16 publications

(27 citation statements)

References 31 publications

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“…Then the set MpL Σ f q of shift-invariant measures on the used symbolic lamination L Σ f :" L Ð Ý Γ f is via (10.3) in natural bijection with the set M f Ď CurrpF N q of currents with support in the used algebraic lamination L FN f . This algebraic lamination is canonically defined by the train track map f , see [36], Definition 3.35 and Lemma 3.36, and it corresponds via (10.4) precisely to the used symbolic lamination L Σ f . We denote by PM f Ď PCurrpF N q the set of projectivized currents rµs defined by any element µ of M f .…”

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confidence: 99%

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Graph towers, laminations and their invariant measures

Bédaride

Hilion

Lustig

2020

J. London Math. Soc.

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In this paper we present a combinatorial machinery, consisting of a graph tower Ð Ý Γ and vector towers Ð Ý v on Ð Ý Γ , which allows us to efficiently describe all invariant measures µ " µ Ð Ý 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 ϕ : F N Ñ F N is determined, when ϕ is represented by a train track map.

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confidence: 99%

“…Proof of Theorem 10.14. From basic train track theory it is known that if a train track map f represents the automorphism ϕ P OutpF N q, then one has ϕ Λ pL FN f q Ď L FN f (see for instance [36]). Furthermore, the algebraic lamination L FN f contains only finitely many sublaminations, each given by a stratum of the expanding train track map f (see [5]).…”

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confidence: 99%

Graph towers, laminations and their invariant measures

Bédaride

Hilion

Lustig

2020

J. London Math. Soc.

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“…The unstable tree is the limit of the sequence T.Ψ −n in CV n for any free simplicial tree T in CV n (see [BFH97] for detailed definition). By a result of [KL14], if Λ + Ψ is the attracting lamination associated to Ψ (as given by Lemma 2.1), then…”

Section: Chl Laminations In [Chl08a]mentioning

confidence: 99%

Loxodromic elements for the relative free factor complex

Gupta

2017

Geom Dedicata

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“…Before explaining this example, we briefly recall some facts related to the index theory of free group automorphisms. We refer the reader to [CH1,KL4,CH2,CHR] for more details.…”

Section: Fibers Of the Cannon-thurston Mapmentioning

confidence: 99%

Cannon–Thurston maps for hyperbolic free group extensions

2016

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This paper gives a detailed analysis of the Cannon-Thurston maps associated to a general class of hyperbolic free group extensions. Let F denote a free groups of finite rank at least 3 and consider a convex cocompact subgroup Γ ≤ Out(F), i.e. one for which the orbit map from Γ into the free factor complex of F is a quasi-isometric embedding. The subgroup Γ determines an extension E Γ of F, and the main theorem of Dowdall-Taylor [DT1] states that in this situation E Γ is hyperbolic if and only if Γ is purely atoroidal.Here, we give an explicit geometric description of the Cannon-Thurston maps ∂F → ∂E Γ for these hyperbolic free group extensions, the existence of which follows from a general result of Mitra. In particular, we obtain a uniform bound on the multiplicity of the Cannon-Thurston map, showing that this map has multiplicity at most 2 rank(F). This theorem generalizes the main result of Kapovich and Lustig [KL5] which treats the special case where Γ is infinite cyclic. We also answer a question of Mahan Mitra by producing an explicit example of a hyperbolic free group extension for which the natural map from the boundary of Γ to the space of laminations of the free group (with the Chabauty topology) is not continuous.

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Invariant Laminations for Irreducible Automorphisms of Free Groups

Cited by 16 publications

References 31 publications

Graph towers, laminations and their invariant measures

Graph towers, laminations and their invariant measures

Loxodromic elements for the relative free factor complex

Cannon–Thurston maps for hyperbolic free group extensions

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