Out(F3) Index Realization

Pfaff, Catherine

doi:10.1017/s0305004115000456

Cited by 9 publications

(5 citation statements)

References 31 publications

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“…Examples for G ϕ with only one F N -orbit of singular points S of ϕ-type, with deg(S) = 3 have been worked out by C. Pfaff, for the case N = 3 (see [54]).…”

Section: The Fibers Of the Cannon-thurston Mapmentioning

confidence: 99%

Cannon-Thurston fibers for iwip automorphisms of FN

Kapovich¹,

Lustig

2015

Journal of the London Mathematical Society

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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 ι : ∂FN → ∂Gϕ.We prove that for any ϕ as above, the map ι is finite-to-one and that the preimage of every point of ∂Gϕ has cardinality ≤ 2N .We also prove that every point S ∈ ∂Gϕ with ≥ 3 preimages in ∂FN has the form (wt m ) ∞ where w ∈ FN , m = 0, and that there are at most 4N − 5 distinct FN -orbits of such singular points in ∂Gϕ (for the translation action of FN on ∂Gϕ).By contrast, we show that for k = 1, 2 there are uncountably many points S ∈ ∂Gϕ (and thus uncountably many FN -orbits of such S) with exactly k preimages in ∂FN .If G is a word-hyperbolic group and H a word-hyperbolic subgroup, and if the inclusion ι : H → G extends to a continuous map ι : ∂H → ∂G, then the map ι is called the Cannon-Thurston map; in this context this definition is due to Mitra [45,46,47]. In particular, if the Cannon-Thurston map ι : ∂H → ∂G exists, then this map is unique and for any sequence h n ∈ H ∪ ∂H converging to some X ∈ ∂H in the topology of H ∪ ∂H, we have lim n→∞ h n = ι(X) in G ∪ ∂G. It is known, see [30, Proposition 2.12], that if H is a non-elementary word-hyperbolic subgroup of a word-hyperbolic group G, then a map ∂H → ∂G is the Cannon-Thurston map if and only if this map is continuous and H-equivariant.

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“…Examples for G ϕ with only one F N -orbit of singular points S of ϕ-type, with deg(S) = 3 have been worked out by C. Pfaff, for the case N = 3 (see [54]).…”

Section: The Fibers Of the Cannon-thurston Mapmentioning

confidence: 99%

Cannon-Thurston fibers for iwip automorphisms of FN

Kapovich¹,

Lustig

2015

Journal of the London Mathematical Society

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“…For N = 3 the statement of the theorem had already been proved by C. Pfaff [11]. Other special cases were also known, for example the single element list [N − 3 2 ] for any N ≥ 3 (see [6]).…”

Section: Introductionmentioning

confidence: 88%

Index realization for automorphisms of free groups

Coulbois¹,

Lustig²

2015

Illinois J. Math.

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For any surface Σ of genus g ≥ 1 and (essentially) any collection of positive integers i 1 , i 2 , . . . , i with i 1 + • • • + i = 4g − 4 Masur and Smillie [10] have shown that there exists a pseudo-Anosov homeomorphism h : Σ → Σ with precisely singularities S 1 , . . . , S in its stable foliation L, such that L has precisely i k + 2 separatrices raying out from each S k .In this paper we prove the analogue of this result for automorphisms of a free group F N , where "pseudo-Anosov homeomorphism" is replaced by "fully irreducible automorphism" and the Gauss-Bonnet equality i 1 +

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“…One can show F has no PNPs using the sage train track package of Coulbois ([Cou14]) or the methodology of [Pfa13], [HM11, Example 3.4], [Pfa15b], [KP18,Lemma 4.7].…”

Section: Main Proofmentioning

confidence: 99%

Taking the High-Edge Route Through Outer Space in Rank 3

Gagnier¹,

Pfaff²

2020

Preprint

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Principal fully irreducible outer automorphisms were introduced in [AKKP19] to emulate principal pseudo-Anosov surface homeomorphisms, i.e. those whose attracting and repelling invariant foliations have only 3-pronged singularities. It is proved in [AKKP19] that these outer automorphisms semi-mimic principal pseudo-Anosov surface homeomorphisms in important ways. We prove here which rank-3 graphs carry train track representatives for principal fully irreducible outer automorphisms. As a corollary, one obtains which simplices of rank-3 Culler-Vogtmann Outer space are passed through by principal axes.

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Out(F₃) Index Realization

Cited by 9 publications

References 31 publications

Cannon-Thurston fibers for iwip automorphisms of FN

Cannon-Thurston fibers for iwip automorphisms of FN

Index realization for automorphisms of free groups

Taking the High-Edge Route Through Outer Space in Rank 3

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