A PDF section boundary and a reverse-stretch diagnostic for Outer Space

Definitions and mathematical statement

The canonical field is a section-boundary fragment, not an independent mathematical question. The aid identifies "Problem 1.1?" as the missing tail of Problem 1.2, followed by the heading Geodesics in Outer Space and an introduction to Section 2. The text attributes to Juan Souto a discussion comparing Teichmüller and Weil--Petersson geodesic analogies with Lipschitz geodesics in Outer Space, and announces a multipart question that is not contained here. The actual Problem 2.1 begins in another record. No missing parts are invented.

The aid gives a distinct mathematical supplement about reverse costs. Let $d$ be an asymmetric metric, let $\gamma:I\to X$ satisfy $d(\gamma(s),\gamma(t))=t-s$ for $s<t$, and define \[ d_{\max}(x,y)=\max\{d(x,y),d(y,x)\},\qquad d_{\Sigma}(x,y)=d(x,y)+d(y,x). \] A quasi-geodesic for one of these symmetrizations means constants $K\ge1,C\ge0$ give the usual two-sided linear distance bounds in $|t-s|$. The aid's equivalence is that this unit forward path is a quasi-geodesic for either symmetrization exactly when \[ \exists A,B\ge0\ \forall s<t\text{ in }I,\qquad d(\gamma(t),\gamma(s))\le A(t-s)+B. \] The forward distance already gives the lower quasi-geodesic bound; the reverse-cost condition supplies the missing upper bound. Constants must be uniform over all pairs of times.

For volume-one marked Outer Space $\mathrm{CV}_N$, $N\ge2$, points are finite marked metric graphs with fundamental group $F_N$. Let $\ell_X(\alpha)$ be the length of a conjugacy class in $X$, and let $d_L(X,Y)$ be the logarithm of the optimal marking-preserving Lipschitz constant. The finite-candidate length-ratio formula gives, with $X=\gamma(s)$, $Y=\gamma(t)$, \[ d_L(Y,X)=\log\max_{\alpha\in\operatorname{Cand}(Y)}\frac{\ell_X(\alpha)}{\ell_Y(\alpha)}. \] The finite source-graph candidate family consists of the standard simple-loop, figure-eight and barbell types. Thus reverse-cost control is equivalent to \[ \ell_{\gamma(s)}(\alpha)\le e^{A(t-s)+B}\ell_{\gamma(t)}(\alpha) \quad\text{for every }\alpha\in\operatorname{Cand}(\gamma(t)),\quad s<t. \] Candidates are taken at the later endpoint, which is the source of the reverse comparison, not automatically at the earlier endpoint.

The aid's sufficient budget criterion combines thick-part quasi-symmetry with control on thin pieces. On a fixed thick part, shortest nontrivial loop lengths have a uniform positive lower bound and known quasi-symmetry bounds reverse cost by a uniform multiple of forward duration. If each interval of the path can be subdivided so that the remaining thin pieces have aggregate reverse cost at most $A(t-s)+B$, uniformly over intervals, the asymmetric triangle inequality gives a linear global reverse budget. An additive allowance on each separate thin piece without controlling their aggregate is not the same hypothesis.

The rank-two thin-rose warning is explicit in compatible coordinates: take the same marking on roses with loop lengths $(1/2,1/2)$ and $(\varepsilon,1-\varepsilon)$, $0<\varepsilon<1/2$. Their forward distance from the balanced rose to the thin rose is $\log(2(1-\varepsilon))<\log2$, while the reverse distance is $\log(1/(2\varepsilon))\to\infty$. Bounded forward duration alone therefore cannot provide uniform reverse control. This supplement is not the missing multipart source problem.

The registry labels the entry open/open triage; attempt 1 is context only, with problem status not a problem, internal rigor proved and novelty confidence low. The aid is machine generated, not peer reviewed, and requires independent verification before citation.

Short English statement

The stored text is an introduction at a PDF section boundary, with no complete question. The aid's separate supplement identifies the extra condition needed for a forward Lipschitz geodesic to remain a quasi-geodesic after symmetrization: a uniform linear reverse-cost bound, equivalently exponential reverse candidate-length bounds. It does not reconstruct the absent multipart problem.

Sources

• Local UnsolvedMath registry, permanent ID $20001641$, source code catalogue entry; aim-geometric-group-theory-notes.json, notes[149].
• AIM, The geometry of the outer automorphism group of a free group, boundary of Problems 1.2 and Section 2, Geodesics in Outer Space, discussion attributed to Juan Souto: https://aimath.org/WWN/outerauto/outerauto.pdf. The boundary identification is the aid's report, not independent PDF inspection.
• Supplied research attempt 1 and registry update August 14, 2026. The original literature field is empty; no full authoritative references for the candidate formula or thick-part quasi-symmetry are supplied.