Thin Kontsevich–Zorich Monodromy Groups

Definitions and mathematical statement

A square-tiled surface is a compact connected translation surface $(X,\omega)$ obtained by gluing finitely many unit Euclidean squares along parallel edges by translations. Equivalently, it admits a finite translation covering of ${\mathbb{C}}/({\mathbb{Z}}+i{\mathbb{Z}})$, branched over at most the image of $0$. Here $\omega$ is a nonzero holomorphic one-form on $X$. Let $\mathcal S_3$ denote the collection of such surfaces with $g(X)=3$; an equivalence relation and a counting convention for this collection have not been specified by the supplied record.

A cylinder in a direction is a maximal open Euclidean cylinder foliated by parallel closed regular geodesics in that direction. A one-cylinder decomposition means that a single such cylinder fills the surface apart from its boundary saddle connections. Under the interpretation that the exclusion concerns every direction, put \[ \mathcal S_3^{\mathrm{no\,1c}} =\{(X,\omega)\in\mathcal S_3:\text{no direction admits a one-cylinder decomposition}\}. \] The record does not explicitly distinguish this interpretation from exclusion in a designated direction, so the directional quantifier requires confirmation.

Kontsevich--Zorich monodromy is the homological monodromy associated with the Gauss--Manin local system over the relevant orbit space; for a square-tiled surface, one considers its associated Teichmüller curve, with orbifold monodromy or a suitable finite cover. Write $\Gamma_X\leq\operatorname{GL}(L_X)$ for the image on a specified invariant integral homology lattice $L_X$. The supplied record does not specify this representation. In particular, full homology $H_1(X,{\mathbb{Z}})$ has rank $6$ in genus $3$, whereas the reduced lattice \[ H_1^{(0)}(X,{\mathbb{Z}}) =\ker\left(p_*:H_1(X,{\mathbb{Z}})\longrightarrow H_1({\mathbb{C}}/({\mathbb{Z}}+i{\mathbb{Z}}),{\mathbb{Z}})\right) \] for a square-tiled covering $p$ has rank $4$. These give distinct possible interpretations of the monodromy in the question; neither is selected here.

For a specified algebraic group $G\leq\operatorname{GL}(L\otimes_{{\mathbb{Z}}}{\mathbb{Q}})$ defined over ${\mathbb{Q}}$, a subgroup $\Gamma\leq G({\mathbb{Q}})\cap\operatorname{GL}(L)$ is called thin relative to $G$ and $L$ if it is Zariski dense in $G$ and \[ \left[G({\mathbb{Q}})\cap\operatorname{GL}(L):\Gamma\right]=\infty. \] One convention takes $G$ to be the Zariski closure of $\Gamma$; another possible intended target is a specified ambient symplectic group. The record supplies no ambient group or convention.

Problem, with unresolved choices displayed. Determine how frequently Kontsevich--Zorich monodromy groups are thin, with particular attention to genus-three square-tiled surfaces. In particular, after fixing the homological representation, lattice, ambient algebraic group, and the directional interpretation above, determine whether \[ \exists (X,\omega)\in\mathcal S_3^{\mathrm{no\,1c}} \quad\text{such that}\quad \Gamma_X\text{ is thin relative to }G_X\text{ and }L_X. \] This formula states the existence target under the every-direction interpretation. It is a conditional formulation, since the missing choices have not been resolved. The broader frequency question also requires a specified population, equivalence relation, size bound, and counting measure or probability measure; no particular density or asymptotic assertion is determined by the source record.

Short English statement

The problem asks how common thin Kontsevich--Zorich monodromy is, especially for square-tiled surfaces of genus $3$, and whether an example can occur when there is no one-cylinder decomposition. A precise existence statement requires specifying the homology representation, the ambient group used to define thinness, and whether the cylinder exclusion applies in every direction. A precise frequency statement additionally requires a counting or measure convention. These choices remain unresolved in the supplied record.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003964$, title Thin Kontsevich--Zorich Monodromy Groups, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The record gives proposed year $2018$ and supplies no named proposer.
• Supplied report citation: Flat Surfaces and Algebraic Curves (2019), Oberwolfach Reports, pp. 2583--2650, DOI 10.4171/owr/2018/42. Supplied source link and literature-source label "Oberwolfach Report 42, thin monodromy": https://doi.org/10.4171/owr/2018/42. The citation's parenthetical year $2019$, the proposed year $2018$, and the DOI's $2018$ component are retained as supplied; their bibliographic relationship has not been verified.
• Supplied literature source, labeled "Hubert--Matheus, genus-three arithmetic monodromy": https://arxiv.org/abs/1806.09855.
• Supplied literature source, labeled "Bonnafoux et al., arithmetic KZ monodromies": https://arxiv.org/abs/2206.06595.
• The supplied literature assessment is dated exactly 2026-08-21. It labels the problem open and reports that genus-three arithmetic examples and families are known, while the frequency and existence of thin Kontsevich--Zorich monodromies in the general genus-three problem remain unresolved. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The supplied background expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that its clean statement is self-contained and preserves the target. These are supplied editorial claims; they do not resolve the formulation choices identified here or establish mathematical correctness. No external source was visited for this draft.