By Michael Handel, Lee Mosher

The authors strengthen a suggestion of axis within the Culler-Vogtmann outer house $\mathcal{X}_r$ of a finite rank loose crew $F_r$, with recognize to the motion of a nongeometric, absolutely 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 usual metric, and $\phi$ turns out to not have a unmarried average axis. in its place those axes for $\phi$, whereas now not distinct, healthy into an ""axis bundle"" $\mathcal{A}_\phi$ with great topological homes: $\mathcal{A}_\phi$ is a closed subset of $\mathcal{X}_r$ right homotopy resembling a line, it really is 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 house, and $\mathcal{A}_\phi$ relies clearly at the repeller and attractor.

The authors suggest quite a few definitions for $\mathcal{A}_\phi$, each one prompted in numerous methods by way of educate tune idea or through houses of axes in Teichmuller house, and so they end up their equivalence.

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If φ is rotationless then φ is geometric if and only if φ is represented by a principle automorphism with nontrivial fixed subgroup. Let g : G → G be an irreducible train track representative of φ. A lift g˘ : G → G is a principal lift if the corresponding automorphism is a principal automorphism, and so two principal lifts g˘, g˘ are equivalent if and only if g˘ = tγ ◦ g˘ ◦ t−1 γ for some γ ∈ Fr . Consider a periodic vertex v ∈ G. Since v has at least one legal turn, v has at least two periodic directions.

A path σ is a periodic Nielsen path if g# minimal p ≥ 1 called the period of σ. A Nielsen path is a periodic Nielsen path with period one. A periodic Nielsen path is indivisible if it can not be written as a non-trivial concatenation of periodic Nielsen subpaths. A path σ is a periodic k (σ) is a periodic pre-Nielsen path if it is not a periodic Nielsen path but some g# Nielsen path. We record information about Nielsen paths in the following lemma. 7. Assume that g : G → G is an irreducible train track map.

The lift fixed, vertex. There is a path σ connecting aj to c so that g# σ ˜ with endpoint a ˜j terminates at a principal vertex c˜ and fg (˜ c) = fg (˜ aj ) = b. This completes the proof of (4) and so the proof of the lemma. 4 of [BFH97], and whose second items follows from the elementary fact that the translation lengths of nontrivial elements of a finite rank free group acting discretely on an R-tree are bounded away from zero. 17. 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.

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