Good Curve Graphs with Unbounded Mapping-Class Orbits

Definitions and mathematical statement

The supplied record does not specify the class $\mathscr S$ of surfaces under consideration or define the invariant $f(S)$. Accordingly, the following is a conditional formulation of the recorded problem, pending clarification of these data.

For $S\in\mathscr S$, let $\operatorname{Mod}(S)$ denote the intended mapping class group. Its orientation and boundary conventions remain unspecified. A good curve graph, in the terminology of the record, is a nonempty connected graph $G$ whose vertices are essential simple closed curves on $S$, equipped with an action of $\operatorname{Mod}(S)$ by graph automorphisms with unbounded orbits. Give each edge length one and write $d_G$ for the resulting path metric. The unboundedness condition can be expressed as \[ \exists v\in V(G)\quad \sup_{g,h\in\operatorname{Mod}(S)}d_G(g\cdot v,h\cdot v)=\infty. \] For an action by graph automorphisms on a connected graph, this condition is equivalent to unboundedness of every vertex orbit.

The source does not settle whether vertices are embedded curves or isotopy classes, which curves count as essential, whether all eligible curves must occur as vertices, or whether the action must be induced by the action on curves. These choices are part of the unresolved definition of the admissible graphs; they must be fixed before the following predicate becomes fully specified: \[ P(S)\;:\Longleftrightarrow\; \text{there exists a good curve graph for }S. \]

Problem. Determine which surfaces $S\in\mathscr S$ satisfy $P(S)$. The supplied conjecture asserts \[ \forall S\in\mathscr S,\qquad f(S)=1\Longrightarrow\neg P(S), \] and predicts both positive and negative examples for $f(S)=2$ or $3$. Reading this prediction separately at each value gives \[ \forall k\in\{2,3\}\quad \exists S_k^+,S_k^-\in\mathscr S:\qquad f(S_k^+)=f(S_k^-)=k,\quad P(S_k^+)\ \text{and}\ \neg P(S_k^-). \] This separate-at-each-value reading requires confirmation from the source. The supplied record calls $f(S)$ a source complexity but gives no definition or explicit identification with the finite-invariance index mentioned in its literature assessment; no such identification is assumed here.

Short English statement

Determine which surfaces admit a connected graph of essential simple closed curves with unbounded mapping-class-group orbits. The recorded conjecture predicts nonexistence when $f(S)=1$ and both existence and nonexistence examples at values $2$ and $3$. The surface class, invariant $f$, and precise graph conventions require clarification before this becomes a fully specified mathematical statement.

Sources

• Local problem registry: permanent UnsolvedMath ID $30003942$, title Good Curve Graphs with Unbounded Mapping-Class Orbits, exact source problem code catalogue entry. The supplied record gives proposed year 2018 and names no proposer.
• Supplied workshop provenance: 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 describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The citation is supplied provenance; its contents have not been independently inspected here.
• Supplied literature reference: Graphs of curves on infinite-type surfaces with mapping class group actions. https://doi.org/10.5802/aif.3217.
• Supplied literature reference: Curve graphs of surfaces with finite-invariance index 1. https://arxiv.org/abs/2106.14916.
• Supplied literature assessment, checked on 2026-08-21: status partially_solved. The assessment states that good curve graphs exist for finite-invariance index at least four and do not exist for index zero, while the index-one, index-two, and index-three cases are not completely classified. This is supplied dated curation derived from the Oberwolfach Reports record, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.