Question 1.2 — Develop a metric theory of Outer space.

Definitions and mathematical statement

For $n\geq2$, Culler--Vogtmann Outer space in its usual projectivized model consists of finite connected rank-$n$ core metric graphs of total edge length $1$, with positive edge lengths and a homotopy-equivalence marking by the rank-$n$ rose, modulo marking-preserving isometry. It carries the $\operatorname{Out}(F_n)$-action. The problem asks to develop a metric theory of Outer space; the source does not prescribe its variant or a symmetric/asymmetric metric.

An iwip, or fully irreducible outer automorphism, has no positive power preserving the conjugacy class of a proper nontrivial free factor. The accompanying discussion seeks an axis theory for such elements, motivated by axial semisimple integral-matrix actions on the source's symmetric space $X$ and unique pseudo-Anosov axes in Teichmüller space. It reports that iwips and pseudo-Anosovs have two distinguished natural-boundary fixed points with north--south dynamics for their cyclic actions [62, 54]. The exact compactifications and metric hypotheses are not supplied.

The source says initial Handel--Mosher work [43] rules out expecting a unique metric axis for iwips, but later work [42] directly constructs an axis bundle for each iwip, with comparison [63]. Thus it retains the broader metric-theory problem despite that progress, rather than asking only for an axis already supplied by the later construction. The claimed matrix analogy, including $X=\mathbb H^2$ in rank two, is contextual and does not specify a chosen Outer-space metric or rank-one property in higher rank.

The extract ends by announcing a modified question, without printing it. Therefore the desired specific remaining metric properties, weaker-axis axioms or quantitative bounds are not recoverable. No conjectured relative hyperbolicity or unique-axis theorem is imposed.

Short English statement

Develop an Outer-space metric and axis theory compatible with fully irreducible dynamics, retaining Handel--Mosher axis-bundle progress and the missing final formulation.

Sources

• Problems on Mapping Class Groups and Related Topics, edited by Farb, Question 1.2 and surrounding metric-theory discussion, supplied PDF page 328. https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. Recorded access 2026-07-29; extraction pdf-font-structure.
• Handel and Mosher [43] and [42], with [63] for axis-bundle comparison; [62], [54] for the boundary-dynamics discussion. Complete bibliography entries are absent.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $11000267$, exact source code catalogue entry. Its OPEN-TRIAGE summary mentions Handel--Mosher and relative-hyperbolicity work without a precise metric, model or reference.