Mapping-Class Orbit Counting in the Thurston Metric

Definitions and mathematical statement

Let $S$ denote the underlying surface, $\mathcal T(S)$ its Teichmüller space of marked conformal structures, and $\operatorname{Mod}(S)$ its mapping class group, acting by changing markings. The supplied record does not specify the topological type of $S$, boundary conditions, or puncture conventions. The following formulation is conditional on choosing these data so that the standard hyperbolic-length definition of the Thurston metric below applies; it does not resolve those missing hypotheses.

Let $\mathcal C(S)$ be the set of free homotopy classes of essential, nonperipheral simple closed curves on $S$. For $z\in\mathcal T(S)$ and $\alpha\in\mathcal C(S)$, write $\ell_z(\alpha)$ for the length of the geodesic representative in the hyperbolic structure associated with $z$. In the usual setting where this formula defines the Thurston metric, put \[ d_{\mathrm{Th}}(u,v) =\log\sup_{\alpha\in\mathcal C(S)} \frac{\ell_v(\alpha)}{\ell_u(\alpha)}. \] This metric is generally asymmetric. Consequently, the phrase "ball about $x$" requires a direction convention. For closed balls, the two corresponding counting functions are \[ \begin{aligned} N^+_{x,y}(R) &=\#\{z\in\operatorname{Mod}(S)\cdot y: d_{\mathrm{Th}}(x,z)\le R\},\\ N^-_{x,y}(R) &=\#\{z\in\operatorname{Mod}(S)\cdot y: d_{\mathrm{Th}}(z,x)\le R\}, \qquad R\ge0. \end{aligned} \] These count distinct orbit points, with no multiplicity from the stabilizer of $y$. In particular, the forward condition is \[ \ell_z(\alpha)\le e^R\ell_x(\alpha) \quad\text{for every }\alpha\in\mathcal C(S). \] The reverse condition instead requires $\ell_x(\alpha)\le e^R\ell_z(\alpha)$ for every such $\alpha$.

Requested asymptotic problem. For arbitrary fixed $x,y\in\mathcal T(S)$, determine the asymptotic behavior, as $R\to\infty$, of the orbit count for the intended Thurston ball. The supplied wording does not select $N^+_{x,y}$ or $N^-_{x,y}$, and does not specify open versus closed balls. The closed-ball formulas above make the directional alternatives explicit without asserting that the source requests both.

For comparison, the extremal length of $\alpha$ on the conformal surface $z$ is \[ \operatorname{Ext}_z(\alpha) =\sup_{\rho} \frac{L_\rho(\alpha)^2}{\operatorname{Area}_\rho(z)}, \] where $\rho$ ranges over conformal metrics of positive finite area and $L_\rho(\alpha)$ is the infimum of the $\rho$-lengths of curves representing $\alpha$. The source further asks how to relate or unify the Thurston orbit asymptotic with counting by extremal length rather than hyperbolic length. It does not specify the objects counted in that comparison, a chosen curve or collection of curves, the extremal-length functional, or its threshold normalization. Thus this comparison remains an unresolved part of the formulation, rather than a uniquely defined second counting problem.

Short English statement

For fixed points $x,y$ in a Teichmüller space, determine how the number of distinct mapping-class orbit points of $y$ in a Thurston ball about $x$ grows as its radius tends to infinity, and relate this to extremal-length counting. The surface conventions, direction of the asymmetric ball, and precise extremal-length comparison require clarification.

Sources

• Local registry: data/unsolvedmath/problems.json, with supplementary records in catalog/problems.json; supplied permanent UnsolvedMath ID $30003956$, source problem code catalogue entry, title "Mapping-Class Orbit Counting in the Thurston Metric."
• Supplied report citation: New Trends in Teichmüller Theory and Mapping Class Groups (2019), Oberwolfach Reports, pp. 2475--2534, DOI 10.4171/owr/2018/40: https://doi.org/10.4171/owr/2018/40. The record separately gives proposed year 2018; these year fields are retained without reconciling them.
• Supplied literature source labeled "Oberwolfach Report 40, Su problem": https://publications.mfo.de/bitstream/handle/mfo/3662/OWR_2018_40.pdf?isAllowed=y&sequence=1. This label associates the problem with Su; the record's proposer field is empty.
• Supplied literature source labeled "Su, variation of extremal length": https://arxiv.org/abs/1210.0743.
• Supplied literature source labeled "Arana-Herrera, effective mapping class dynamics II": https://link.springer.com/article/10.1007/s00039-026-00740-6.
• Supplied literature assessment, checked 2026-08-21: the record labels the problem open and reports known Teichmüller-metric orbit counting, partial Thurston-metric and extremal-length results, and unproved requested extremal-length asymptotics and unification. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.