By Michael Handel, Lee Mosher

ISBN-10: 0821869272

ISBN-13: 9780821869277

The authors increase a thought of axis within the Culler-Vogtmann outer area $\mathcal{X}_r$ of a finite rank unfastened workforce $F_r$, with appreciate to the motion of a nongeometric, totally irreducible outer automorphism $\phi$. in contrast to the location of a loxodromic isometry performing on hyperbolic house, or a pseudo-Anosov mapping type performing on Teichmuller house, $\mathcal{X}_r$ has no usual metric, and $\phi$ turns out to not have a unmarried ordinary axis. as an alternative those axes for $\phi$, whereas no longer precise, 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 reminiscent of a line, it really is invariant lower 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 certainly at the repeller and attractor.

The authors suggest a variety of definitions for $\mathcal{A}_\phi$, every one stimulated in numerous methods through teach song thought or through homes of axes in Teichmuller house, they usually end up their equivalence.

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**Example text**

7 of [FH06]. The path σ of (2) is a Nielsen path. If R1 and R2 eventually coincide then ˜ ⊂ R1 and β˜ ⊂ R2 . This proves the only if part of (2). For the σ ˜=α ˜ β˜−1 where α if part, suppose that σ = αβ¯ is indivisible. Let α ˜ and β˜ be the lifts of α and β that begin at v˜1 and v˜2 respectively. For all n ≥ 1, there are paths τ˜n such that ˜ = β˜τ˜n . The paths g˘n (α) ˜ = α˜ ˜ τn and g˘n (β) ˜ are an increasing sequence whose g˘n (α) ˜ are an increasing sequence whose union is R2 . union is R1 and the paths g˘n (β) This completes the proof of (2).

8. Relating Λ− to T− and to T+ . Consider an R-tree T representing a point in ∂Xr , and a nonempty minimal sublamination Λ of GFr . Following [BFH97] section 3 we say that Λ has length zero in T if for every G ∈ Xr and every Fr equivariant morphism h : G → T there exists C ≥ 0 such that for every leaf ⊂ G of ΛG we have diamT− (h( )) ≤ C. 20. For each fully irreducible φ ∈ Out(Fr ), Λ− = Λ− (φ) is the unique minimal lamination that has length zero in T− = T− (φ). Proof. 5 (4) of [BFH97], but restricted to the so-called “irreducible laminations”, meaning the set of expanding laminations of all fully irreducible outer automorphisms in Fr .

While we apply these concepts here only to fully irreducible outer automorphisms, more general deﬁnitions are given in [FH06] which apply to arbitrary outer automorphisms and their relative train track representatives. Let φ ∈ Out(Fr ) be fully irreducible, consider Φ ∈ Aut(Fr ) representing φ, and ˆ : ∂Fr → ∂Fr . Denote the ﬁxed point set consider also the boundary extension Φ ˆ by Fix(Φ), ˆ and denote the subset of nonrepelling ﬁxed points of Fix(Φ) ˆ by of Φ 2. PRELIMINARIES 27 ˆ We say that Φ is a principal automorphism if FixN (Φ) ˆ contains at least FixN (Φ).

### Axes in outer space by Michael Handel, Lee Mosher

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