Lone axes in outer space

Mosher, Lee; Pfaff, Catherine

doi:10.2140/agt.2016.16.3385

Cited by 13 publications

(19 citation statements)

References 22 publications

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“…We take the occasion of the present discussion on train tracks for free product to give explicit statements and metric proofs of some of these facts, as for instance the relations between the minimal displaced set and the set of train tracks. (See also [22,32]).…”

Section: Train Tracksmentioning

confidence: 99%

“…As the set of minimally displaced elements is closed, this gives in particular a proof that the set of train tracks is closed, hence answering to a question raised in [22], where the authors give a characterizations of the axis bundle of an irreducible automorphism (see Remark 8.22). We would also like to mention the very recent preprint [32] about axis bundles.…”

Section: Introductionmentioning

confidence: 99%

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Stretching factors, metrics and train tracks for free products

Francaviglia¹,

Martino²

2015

Illinois J. Math.

128

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In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same as in the classical case, with some technicalities arising from the presence of infinite-valence vertices.We describe the Lipschitz metric and show how to compute it; we prove the existence of optimal maps; we describe geodesics represented by folding paths.We show that train tracks representative of irreducible (hence hyperbolic) automorphisms exist and that their are metrically characterized as minimal displaced points, showing in particular that the set of train tracks is closed (in particular answering to some questions raised in [22] concerning the axis bundle of irreducible automorphisms).Finally, we include a proof of the existence of simplicial train tracks map without using Perron-Frobenius theory.A direct corollary of this general viewpoint is an easy proof that relative train track maps exist in both the free group and free product case.

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Section: Train Tracksmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Stretching factors, metrics and train tracks for free products

Francaviglia¹,

Martino²

2015

Illinois J. Math.

128

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“…More recently Mosher and Pfaff [MP13] proved a necessary and sufficient ideal Whitehead graph condition for an axis bundle to be a unique, single axis. The essence of this fact is that the axis bundle for a fully irreducible φ is the closure of the set of points in outer space (marked graphs with lengths on edges) on which there exists an affine train track representative for some power φ k of φ.…”

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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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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“…In particular, the tools will allow us to find a fully irreducible in each rank whose ideal Whitehead graph has a cut vertex. By [MP13], this gives an example in each rank of a fully irreducibles whose axis bundle is more than just a single axis. It further gives examples of "nongeneric" behavior in the sense of [KP15].…”

Section: Introductionmentioning

confidence: 99%

“…One important fact about the axis bundle is that, if ϕ and ψ are two fully irreducible outer automorphisms, then A ϕ and A ψ differ by the action of an Out(F r ) on its Culler-Vogtmann outer space if and only if ϕ and ψ have powers conjugate in Out(F r ). More recently Mosher and Pfaff [MP13] proved a necessary and sufficient ideal Whitehead graph condition for an axis bundle to be a unique, single axis. [Pfa13] gives examples in each rank where this occurs.…”

Section: Introductionmentioning

confidence: 99%

Ideal Whitehead Graphs in Out(F r )

Pfaff

2014

Trends in Mathematics

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We provide an example in each rank of an ageometric fully irreducible outer automorphism whose ideal Whitehead graph has a cut vertex. Consequently, we show that there exist examples in each rank of axis bundles (in the sense of [HM11]) that are not just a single axis, as well as of "nongeneric" behavior in the sense of [KP15]. The tools developed here also allow one to construct a whole new array of ideal Whitehead graphs achieved by ageometric fully irreducible outer automorphisms in all ranks.

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Lone axes in outer space

Cited by 13 publications

References 22 publications

Stretching factors, metrics and train tracks for free products

Stretching factors, metrics and train tracks for free products

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

Ideal Whitehead Graphs in Out(F r )

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