Research Problems in Function Theory — Problem 7.83

Definitions and mathematical statement

Let $G\le\operatorname{PSL}(2,{\mathbb{R}})$ be a finitely generated Fuchsian group of the first kind: it is discrete and its limit set is the whole ideal boundary of the hyperbolic plane. The source asks two questions.

1. If $F:\Pi\to\Pi$ is a $G$-invariant quasisymmetric function, must $F$ be totally singular?
2. Is the Teichmüller space $T_G$ dense in $S_G$, described as the Schwarzians of $G$-invariant univalent functions?

The extract does not define $\Pi$, either meaning of $G$-invariant, or the topology in the second question. These are essential mathematical data.

In a boundary-homeomorphism interpretation, quasisymmetry controls distortion of ratios of neighboring boundary intervals, and total singularity ordinarily refers to a singular boundary measure or the appropriate derivative-zero-almost-everywhere condition. But the boundary coordinate, measure, and precise singularity convention are absent. Literal invariance $F\circ g=F$ is incompatible with an injective $F$ for a nonidentity $g$; equivariance, conjugating $G$ to another Fuchsian group, and invariance of an extension's Beltrami coefficient are different possible intended notions. No exception for a trivial or Möbius deformation is stated. Thus no unconditional singularity theorem is inferred from the printed wording.

For a locally univalent holomorphic map $f$, the Schwarzian derivative is \[ \mathcal S(f)=\frac{f^{(3)}}{f'}-\frac32\left(\frac{f^{(2)}}{f'}\right)^2. \] The density question needs the function domain, invariant or equivariant condition, a realization $\iota:T_G\to S_G$, and a topology $\tau$. Its target would then be \[ \overline{\iota(T_G)}^{\,\tau}=S_G. \] The source does not choose a norm topology, locally uniform convergence, or another topology. Literal invariance of a univalent function has the same injectivity issue; automorphy of its Schwarzian is different and is not silently substituted.

The supplied triage labels both questions open as of Hayman's 2018 edition and reports no definitive later resolution. This is an attributed source assessment, not an independent literature check or a claim that any particular repaired formulation remains open.

Short English statement

For a finitely generated first-kind Fuchsian group, the source asks about singularity of invariant quasisymmetric maps and density of its Teichmüller space among specified Schwarzians. The boundary symbol, invariance, singularity, embedding, and topology are undefined. They must be recovered before either question has a unique precise formulation.

Sources

• Local source registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $2307083$, source code catalogue entry, including the supplied OPEN-TRIAGE assessment. Dataset attribution: Ulam AI and UnsolvedMath Contributors; original-source attribution and rights are retained.
• Hayman, Research Problems in Function Theory, source-list date 2018, Problem 7.83, attributed collectively to O. Lehto, L. Bers, and I. Kra, communicated by D. Hamilton: https://arxiv.org/abs/1809.07200. The metadata reports source-TeX extraction and July 29, 2026 access, with status and rights review flags. Both questions are retained despite corrupted list markers.
• No separate authoritative reference, definition of $\Pi$, invariance convention, or density topology is supplied. No linked source was inspected for this draft.