Kirby Problem 3.39

Definitions and mathematical statement

The weight $w(G)$ of a finitely generated group is the smallest size of a set $A\subseteq G$ whose normal closure is $G$: \[ G=\langle\!\langle A\rangle\!\rangle =\langle gag^{-1}:g\in G,\ a\in A\rangle. \] Set $w(1)=0$. A group is perfect if $G=[G,G]$, or equivalently its abelianization is trivial. An integral homology three-sphere is a closed connected oriented manifold $Y$ with $H_*(Y;\mathbb Z)\cong H_*(S^3;\mathbb Z)$; its fundamental group is perfect.

The catalogue asks four questions: (a) does every nontrivial finitely generated perfect group have weight one? (b) can an integral homology three-sphere group have weight greater than one? (c) for every $N$, can such a group have weight greater than $N$? (d) are the weights unbounded already for Seifert fibered homology spheres $\Sigma(a_1,\ldots,a_r)$? These are circle-fibered three-manifolds over a sphere with exceptional fiber multiplicities $a_i$, pairwise coprime in the homology-sphere setting.

Status distinction. Chen--Lodha's 2025 preprint gives a negative answer to (a) and constructs hyperbolic integral homology spheres answering (b) positively [S3]. The remaining targets recorded here are the unboundedness questions (c) and (d); the lower bound two from that result does not establish arbitrarily large weight.

Short English statement

Determine how large the normal generating number of an integral homology three-sphere group can be, particularly among Seifert fibered examples.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2837 (KP-3.39), 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 3.39. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.
• S3 L. Chen and Y. Lodha, The Wiegold problem and free products of left-orderable groups, v2 (2025), Theorems A and C. Resolves parts (a) and (b), distinguished from the remaining questions. https://arxiv.org/abs/2510.26073v2.