Finite-Volume Symplectic Quotients of Hitchin Components

Definitions and mathematical statement

Let $\mathcal H$ denote the Hitchin component intended in the source, equipped with its intended symplectic form $\omega$. The supplied record does not identify the underlying surface, the target Lie group, the component, or the symplectic form. Consequently, the formulation below is conditional on specifying these data; it does not quantify universally or existentially over choices of these parameters.

For a symplectic manifold $(\mathcal H,\omega)$ of dimension $2m$, write \[ \operatorname{Symp}(\mathcal H,\omega) =\{f:\mathcal H\longrightarrow\mathcal H\mid f\text{ is a smooth diffeomorphism and }f^*\omega=\omega\}. \] For this draft, a subgroup $\Gamma\leq\operatorname{Symp}(\mathcal H,\omega)$, regarded as a discrete group, acts properly discontinuously if, for every compact subset $K\subseteq\mathcal H$, \[ \#\{\gamma\in\Gamma:\gamma K\cap K\neq\varnothing\}<\infty. \] This convention permits finite stabilizers. Interpret finite volume as finite total measure on $\Gamma\backslash\mathcal H$ induced by the symplectic volume density $\left|\omega^{\wedge m}/m!\right|$, using orbifold volume when stabilizers are nontrivial. If the action is free, the descended symplectic form $\bar\omega$ gives the equivalent condition \[ \int_{\Gamma\backslash\mathcal H}\left|\frac{\bar\omega^{\wedge m}}{m!}\right|<\infty. \]

With these conventions and the intended data fixed, the source question is: does there exist a subgroup $\Gamma\leq\operatorname{Symp}(\mathcal H,\omega)$ whose action is properly discontinuous and whose quotient has finite symplectic volume?

The source does not specify the volume measure, whether the action must be free, or its precise convention for proper discontinuity. The conventions above therefore require confirmation alongside the missing Hitchin-component data. No compactness requirement on the quotient is stated.

Short English statement

The question asks whether the intended Hitchin component admits a properly discontinuous group of symplectic transformations with a quotient of finite volume. The surface, target group, symplectic form, volume convention, and allowance of finite stabilizers require clarification. The supplied registry labels the problem open; this draft does not independently verify that status.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003967$, title Finite-Volume Symplectic Quotients of Hitchin Components, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. No individual proposer is identified. The supplied proposed year is 2018.
• Supplied report citation: Flat Surfaces and Algebraic Curves (2019), Oberwolfach Reports, pp. 2583--2650, DOI 10.4171/owr/2018/42. Source URL and literature-source link labeled "OWR report": https://doi.org/10.4171/owr/2018/42. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature-source link labeled "Hamiltonian flows on character varieties": https://ul.qucosa.de/api/qucosa%3A97838/attachment/ATT-0/. No author, publication date, or more detailed bibliographic identification is supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the report poses it as an open question, while later symplectic and Hamiltonian results do not provide a properly discontinuous finite-volume action on the Hitchin component. This is supplied dated curation, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry supplies the statement-status label corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting that the clean statement is self-contained and preserves the mathematical target. That supplied review assertion does not resolve the missing mathematical specifications identified in this draft.