Research Problems in Function Theory — Problem 5.35

Definitions and mathematical statement

Let $\mathbb D=\{z\in{\mathbb{C}}:|z|<1\}$, and let $\Gamma\le\operatorname{Aut}(\mathbb D)$ be a discontinuous group of Möbius transformations, in the usual discrete disk-automorphism sense. A meromorphic map $f:\mathbb D\to\widehat{\mathbb{C}}$ is automorphic with respect to $\Gamma$ when \[ f(\gamma z)=f(z)\qquad(\gamma\in\Gamma,\ z\in\mathbb D). \] Its spherical derivative is $f^\#(z)=|f'(z)|/(1+|f(z)|^2)$, extended across poles using the coordinate $1/f$. The required normality condition is \[ \sup_{z\in\mathbb D}(1-|z|^2)f^\#(z)<\infty. \] The source asks whether every discontinuous group admits an automorphic meromorphic function satisfying this condition. It first names the group $F$ and later $\Gamma$; the intended identification of these two symbols should be confirmed.

As transcribed, no nonconstancy condition is imposed. Every constant function is automorphic and has spherical derivative zero, so the literal existence statement is automatic. A meaningful intended problem would ask whether, for every such $\Gamma$, there exists a nonconstant meromorphic $f$ satisfying both displayed properties. This is an explicit possible correction, not a silent change to the supplied question. No finite-generation, finite-covolume, torsion-free, or compact-quotient hypothesis is added. The supplied OPEN-TRIAGE summary reports open status in the 2018 edition and no definitive resolution located, without addressing the missing nonconstancy clause.

Short English statement

Does every discontinuous disk Möbius group admit a normal automorphic meromorphic function? Constants make the transcribed statement automatic, so the intended nonconstant formulation and group-symbol mismatch require review.

Sources

• Local UnsolvedMath registry, permanent ID $2305035$, exact source code catalogue entry, "Research Problems in Function Theory --- Problem 5.35."
• Hayman, Research Problems in Function Theory (2018), Problem 5.35, attributed to Ch. Pommerenke; https://arxiv.org/abs/1809.07200. Source-TeX extraction; recorded access 2026-07-29.
• Supplied OPEN-TRIAGE summary reports no definitive resolution, without a separate authoritative current-status reference.