Kourovka Notebook Problem 21.127

Definitions and mathematical statement

Let $\Gamma=(V,E)$ be a finite simple graph. Its right-angled Artin group is \[ A_\Gamma=\langle v\ (v\in V)\mid uv=vu\text{ whenever }\{u,v\}\in E\rangle. \] Thus each graph vertex is a group generator, and the only defining relations are commutation relations for adjacent vertices. Put $G=A_\Gamma$ and use $[x,y]=x^{-1}y^{-1}xy$.

The commutator subgroup $[G,G]$ is generated by all commutators. For $g\in[G,G]$, its commutator length $\operatorname{cl}_G(g)$ is the least integer $r\ge0$ such that $g$ is a product of $r$ commutators of elements of $G$, with $\operatorname{cl}_G(1)=0$. Define the stable commutator length \[ \operatorname{scl}_G(g)=\lim_{n\to\infty} \frac{\operatorname{cl}_G(g^n)}{n} =\inf_{n\ge1}\frac{\operatorname{cl}_G(g^n)}{n}. \] The limit and the equality with the infimum follow from subadditivity of the numerator as a function of $n$.

Is $\operatorname{scl}_G(g)$ rational for every finite graph $\Gamma$ and every $g\in[A_\Gamma,A_\Gamma]$? The question includes zero as a rational value and asks only for rationality, with no common bound on denominators. The commutators in the definition may use arbitrary elements of the whole ambient Artin group. Stable commutator length is therefore taken in $G$, rather than intrinsically in the cyclic subgroup generated by $g$ or in a chosen smaller subgroup.

Short English statement

Is stable commutator length rational on the commutator subgroup of every right-angled Artin group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2636 (KOU-21.127), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.127. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.