Rips induction: index of the dual lamination of an $\mathbb{R}$-tree

Coulbois, Thierry; Hilion, Arnaud

doi:10.4171/ggd/218

Cited by 17 publications

(51 citation statements)

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“…While exploration into understanding ideal Whitehead graphs is still a young endeavour, the index theory for free group outer automorphisms has in some directions already been extensively developed. In fact, there are three types of Out(F r ) index invariants in the literature, those of [GL95], [GJLL98], and [CH10]. The index of φ, as defined and studied in [GJLL98], is equal to the geometric index of T + φ , as established by Gaboriau-Levitt [GL95] for more general R-trees.…”

Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 99%

“…The index of φ, as defined and studied in [GJLL98], is equal to the geometric index of T + φ , as established by Gaboriau-Levitt [GL95] for more general R-trees. [CH12] provides a relationship between the index of [CH10] and the geometric index, as well as uses the index to relate different properties of the attracting and repelling tree for a fully irreducible. There are also even index realization results of several different natures.…”

Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 99%

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Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Pfaff

2016

J. Topol. Anal.

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Abstract. By proving precisely which singularity index lists arise from the pair of invariant foliations for a pseudo-Anosov surface homeomorphism, Masur and Smillie [MS93] determined a Teichmüller flow invariant stratification of the space of quadratic differentials. In this final paper of a three-paper series, we give a first step to an Out(Fr) analog of the Masur-Smillie theorem. Since the ideal Whitehead graphs defined by Handel and Mosher [HM11] give a strictly finer invariant in the analogous Out(Fr) setting, we determine which of the twenty-one connected, simplicial, five-vertex graphs are ideal Whitehead graphs of fully irreducible outer automorphisms in Out(F3). IntroductionLet F r denote the free group of rank r and Out(F r ) its outer automorphism group. In this paper we prove realization results for an invariant dependent only on the conjugacy class (within Out(F r )) of the outer automorphism, namely the "ideal Whitehead graph." 1.1. Main result. A "fully irreducible" (iwip) outer automorphism is the most commonly used analogue to a pseudo-Anosov mapping class and is generic. An element φ ∈ Out(F r ) is fully irreducible if no positive power φ k fixes the conjugacy class of a proper free factor of F r .We give in Subsection 2.3 the exact Out(F r ) definition of an ideal Whitehead graph. For now, to give context, we remark that, for a pseudo-Anosov surface homeomorphism, the component of an ideal Whitehead graph coming from a foliation singularity is a polygon with edges corresponding to the lamination leaf lifts bounding a principal region in the universal cover [NH86].Handel and Mosher define in [HM11] a notion of an ideal Whitehead graph for a fully irreducible outer automorphism, a finite graph whose isomorphism type is an invariant of the conjugacy class of the outer automorphism. In this paper we investigate the extent to which the Out(F r ) situation is more complicated by giving a partial answer to a question posed by Handel and Mosher in [HM11]: Question 1.1. For each r ≥ 2, which isomorphism types of graphs occur as IW(φ) for a fully irreducible φ ∈ Out(F r )? Theorem. A. Exactly eighteen of the twenty-one connected, simplicial five-vertex graphs are the ideal Whitehead graph IW(φ) for a fully irreducible outer automorphism φ ∈ Out(F 3 ).The twenty-one connected, simplicial five-vertex graphs ([CP84]

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Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 99%

Section: An Ideal Whitehead Graph Definitionmentioning

confidence: 99%

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Pfaff

2016

J. Topol. Anal.

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“…Let T be an indecomposable tree. In our previous work [CH14], we proved that there are two cases: Levitt type or surface type. According to this dichotomy, we describe the unfolding induction used in the present paper.…”

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confidence: 97%

“…By our previous work [CH14], the index of is finite and thus it contains a finite core graph ∞ : the union of all reduced loops in . We remark that ∞ can be empty and that it can fail to be connected.…”

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confidence: 99%

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Ergodic currents dual to a real tree

Coulbois¹,

Hilion²

2014

Ergod. Th. Dynam. Sys.

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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 $\mu$ dual to $T$, the unfolding induction produces a sequence of approximations converging towards $\mu$. We also give a unique ergodicity criterion.Comment: 14 pages, minor corrections from previous versio

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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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Rips induction: index of the dual lamination of an $\mathbb{R}$-tree

Cited by 17 publications

References 30 publications

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Ideal Whitehead graphs in Out(Fr) III: Achieved graphs in rank 3

Ergodic currents dual to a real tree

Cannon–Thurston maps for hyperbolic free group extensions

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