By Michael Handel, Lee Mosher

The authors boost a concept of axis within the Culler-Vogtmann outer house $\mathcal{X}_r$ of a finite rank unfastened staff $F_r$, with admire to the motion of a nongeometric, totally irreducible outer automorphism $\phi$. not like the location of a loxodromic isometry performing on hyperbolic area, or a pseudo-Anosov mapping classification performing on Teichmuller house, $\mathcal{X}_r$ has no typical metric, and $\phi$ turns out to not have a unmarried typical axis. as a substitute those axes for $\phi$, whereas no longer designated, healthy into an ""axis bundle"" $\mathcal{A}_\phi$ with great topological houses: $\mathcal{A}_\phi$ is a closed subset of $\mathcal{X}_r$ right homotopy comparable to a line, it's invariant less than $\phi$, the 2 ends of $\mathcal{A}_\phi$ restrict at the repeller and attractor of the source-sink motion of $\phi$ on compactified outer area, and $\mathcal{A}_\phi$ relies obviously at the repeller and attractor.

The authors suggest quite a few definitions for $\mathcal{A}_\phi$, every one encouraged in numerous methods by means of teach tune thought or by means of homes of axes in Teichmuller area, they usually turn out their equivalence.

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For each nongeometric, fully irreducible φ ∈ Out(Fr ) and each nontrivial subgroup A < Fr of infinite index, the following hold: (1) The action of A on T+φ is discrete. (2) The translation lengths of nontrivial elements of A acting on T+φ have a positive lower bound. 6. Geodesic laminations in trees and marked graphs. Given a topological space S let D2 S denote the double space of S, the space of distinct ordered 34 MICHAEL HANDEL AND LEE MOSHER pairs in S: D2 S = {(ξ, η) ∈ S × S ξ = η} The group Z/2 acts freely on D2 S by permuting coordinates.

When s > 0 we say that Gs is obtained from G by a length s fold of e and e , and the quotient map G → Gs is called a fold map from G to Gs . Consider the map γ : [0, S] → Xr defined by γ(s) = Gs . First, γ is a continuous path in Xr from G to GS , and so it descends to a continuous path γ : [0, S] → Xr which, its domain being compact, is proper. Second, for 0 ≤ s ≤ t ≤ S the induced map fts : Gs → Gt is a fold map. Third, the family of maps fts satisfies the train track property and the semiflow property, by construction.

G ˘2  T+ W Φ+ Figure 2. Notation associated to a train track map g : G → G and automorphism Φ representing φ. Maps denoted g˜ depend only on g and are Fr -equivariant. 1). 15. 1), on the other hand the notation g is meant to suggest Fr -equivariance. The following theorem and corollary seem to be folklore. g. 3 below for a proof of a more general fact in the nongeometric case. 13. The direct limit of the direct system gij : Gj → Gi , in the category of metric spaces with an Fr action whose morphisms are Fr equivariant, distance nonincreasing maps, is an Fr tree in the same homothety class as T+ .

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