Problem 21 — (Exceptional Strata).

Definitions and mathematical statement

Write $\mathcal Q(k_1,\ldots,k_s)$ for the relevant moduli stratum of meromorphic quadratic differentials $q$ on compact Riemann surfaces, with prescribed zero/pole orders $k_i$, where order $-1$ is a simple pole. The degree relation is $\sum_i k_i=4g-4$. Thus the listed strata \[ \mathcal Q(-1,9),\quad\mathcal Q(-1,3,6),\quad \mathcal Q(-1,3,3,3),\quad\mathcal Q(12) \] lie in genera $3,3,3,4$, respectively. The precise convention concerning global squares of abelian differentials and labeled singularities must agree with the source; it is not restated in the excerpt.

Problem 21 (exceptional strata). Find a geometric invariant distinguishing the different connected components of each of these four strata. More precisely, for each listed stratum $\mathcal Q$, seek an invariant $I_{\mathcal Q}$ of its differential/flat geometry that is constant on each connected component and takes different values on different components. Equivalently, it should induce an injective labeling of $\pi_0(\mathcal Q)$ by geometrically defined data. The problem is to distinguish components within each fixed stratum, not merely distinguish the four different singularity-order lists.

The source says that the known distinguishing invariant, the extended Rauzy class, is combinatorial rather than geometric, citing [La]. Merely renaming that class is not a geometric construction. The excerpt gives no proposed invariant, number of components, or formal criterion for what geometric description is acceptable.

Short English statement

Find a geometric way to tell apart connected components in the four specified exceptional quadratic-differential strata, where the source currently supplies only extended Rauzy classes as combinatorial labels.

Sources

• Local UnsolvedMath registry, permanent ID 11000228, source code catalogue entry; no proposer or proposal year supplied.
• Farb, edited volume Problems on Mapping Class Groups and Related Topics, Problem 21, PDF page 264; 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.
• Source label [La] for the extended-Rauzy-class component distinction; the complete bibliography entry is absent. The research summary supplies no verified later status.