Kirby Problem 2.33

Definitions and mathematical statement

Fix a compact connected surface $S$ without boundary. Let $\operatorname{Homeo}(S)$ be the group of all self-homeomorphisms of $S$, with composition as its operation. A group $G$ acts on $S$ by homeomorphisms through a homomorphism $\rho:G\to\operatorname{Homeo}(S)$. Its action is faithful precisely when $\ker\rho=\{1\}$.

A group is finitely generated if some finite subset generates it under multiplication and inversion. It is torsion-free if $g^n=1$ for a positive integer $n$ implies $g=1$. The question asks whether, for each fixed $S$, there exists a finitely generated torsion-free group $G$ such that \[ \ker\rho\ne\{1\}\quad \text{for every homomorphism }\rho:G\to\operatorname{Homeo}(S). \] Thus $G$ must admit no faithful topological action on that surface. The group may depend on $S$; the formulation does not require a single group working for all surfaces. No differentiability, measure-preservation, or orientation- preservation assumption is imposed on the actions. In particular, obstructing only smooth actions would not answer the question, and nonfaithful actions are allowed to be nontrivial.

Short English statement

For each closed surface, find a finitely generated torsion-free group that cannot act faithfully on it by homeomorphisms, or show this is impossible.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2781 (KP-2.33), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.33. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.