Question 1.3 — Describe the geometry of the axis bundle (and associated objects) for an iwip acting on Outer Space.

Definitions and mathematical statement

Let $F_n$ be a finite-rank free group, $n\geq2$, and let $\phi\in\operatorname{Out}(F_n)$ be iwip (fully irreducible): no positive power of $\phi$ preserves the conjugacy class of a proper nontrivial free factor. Outer Space $CV_n$ in a standard projectivized model consists of marked finite metric graphs of rank $n$, with positive edge lengths normalized to total length one, modulo marking-compatible isometry; equivalent graph conventions suppress valence-two vertices. The outer automorphism group acts by changing markings.

The source asks to describe the geometry of the axis bundle of $\phi$, together with associated objects. It does not define that bundle or those objects. Common axis-related constructions involve folding lines associated with the attracting and repelling trees in the boundary of Outer Space, while displacement-minimizing loci depend on a chosen metric. These descriptions cannot silently be identified with the source's axis bundle, and no equality between them is asserted here.

A precise geometric answer must first specify the axis-bundle definition, projectivized versus unprojectivized model, topology and metric or folding-path structure, and then state the geometric/topological properties or invariants it describes. The Lipschitz geometry is asymmetric unless a symmetrization is chosen, so "axis" cannot be treated as an ordinary unique metric geodesic without further conditions. The source states iwip generally and does not restrict to atoroidal/ageometric cases; any such restriction needed for a particular construction would have to be made explicit. The rank range and desired completeness criterion for a description are otherwise absent.

Short English statement

Describe the axis-bundle geometry for a fully irreducible free-group outer automorphism. The precise bundle, metric and associated objects are not defined, so a particular folding-line or minimum-displacement model cannot be assumed.

Sources

• Local UnsolvedMath registry, permanent ID $11000268$, exact source code catalogue entry.
• Farb, Problems on Mapping Class Groups and Related Topics, Question 1.3, PDF page 328; supplied running footer M. Bridson and K. Vogtmann: https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Registry access 2026-07-29; pdf-font-structure extraction.
• The summary refers generally to iwip/axis-bundle literature without any precise theorem or reference.