Problem 7.5 — (Shortest Teichm¨ uller loop in a stratum).

Definitions and mathematical statement

Fix an integer $g\ge2$ and let $\operatorname{Mod}_g$ be the mapping class group of a closed orientable genus-$g$ surface. For a pseudo-Anosov mapping class $f$, let $\lambda(f)>1$ denote its stretch factor. With the usual Teichmüller metric normalization, its translation length and associated closed-geodesic length are $\log\lambda(f)$.

The preceding local source item, ID 11000080, defines $(k_1,\ldots,k_r)$ as prong counts at the $r$ singularities of a stable foliation, not orders of zeros of a differential. For a closed unmarked surface, $k_i\ge3$ and the Poincaré--Hopf constraint is \[ \sum_{i=1}^r(k_i-2)=4g-4. \] Call a tuple admissible when it is realizable in the intended foliation setting; this is not an assertion that the displayed numerical condition alone is sufficient. The source attributes pseudo-Anosov realization of every admissible tuple to Masur--Smillie [MS]. No orientability or connected-stratum-component condition is imposed in the question.

Define the minimum-length infimum for those singularity data by \[ \lambda_g(k_1,\ldots,k_r)= \inf\{\log\lambda(f):f\in\operatorname{Mod}_g\text{ pseudo-Anosov, with stable foliation having prong counts }(k_1,\ldots,k_r)\}. \] Although this symbol uses $\lambda$, it denotes a logarithmic length, not the stretch factor itself.

Problem 7.5. For every fixed $g\ge2$ and every admissible tuple, give upper and lower bounds for $\lambda_g(k_1,\ldots,k_r)$ in terms of the genus and singularity data. The source interprets this as bounding the shortest Teichmüller loop in the corresponding substratum in moduli space, obtained from the relevant cotangent-bundle substratum. The defining condition is on the invariant foliation; merely specifying that the projected loop lies in a projection of a stratum can forget this data, so the infimum formula remains the precise target.

Writing \[ L(\operatorname{Mod}_g)=\inf_{f\text{ pseudo-Anosov}}\log\lambda(f), \] the source notes \[ L(\operatorname{Mod}_g)=\min_{\text{admissible }(k_1,\ldots,k_r)} \lambda_g(k_1,\ldots,k_r), \] where only finitely many singularity tuples occur for fixed $g$. The problem concerns bounds for every specified tuple rather than only this overall minimum.

Short English statement

For each admissible stable-foliation prong pattern in genus at least two, bound the infimum of pseudo-Anosov logarithmic stretch factors with exactly that pattern. Taking the minimum over patterns recovers the unrestricted shortest Teichmüller-loop length.

Sources

• Local UnsolvedMath registry, permanent ID 11000081, source code catalogue entry; no individual proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Problem 7.5, PDF page 58; https://www.math.uchicago.edu/~farb/papers/mcgbook.pdf. The registry records PDF-font-structure extraction and access on 2026-07-29; the inventory distinguishes 329 labeled headings by source page.
• Adjacent local ID 11000080 supplies the prong-count convention, finite-tuple observation, and Masur--Smillie realization attribution [MS]. No full bibliography entry for [MS] is supplied here.