Kirby Problem 2.31

Definitions and mathematical statement

Let $S$ be a closed connected oriented surface of genus at least two, and put $G=\pi_1(S)$. Write $[G,G]$ for its commutator subgroup and $[a,b]=aba^{-1}b^{-1}$. For $x\in[G,G]$, define its commutator length by \[ \operatorname{cl}_G(x)=\min\{k\geq0:x=[a_1,b_1]\cdots[a_k,b_k],\ a_i,b_i\in G\}. \] The empty product gives $\operatorname{cl}_G(1)=0$. Its stable commutator length is \[ \operatorname{scl}_G(x)=\lim_{n\to\infty} \frac{\operatorname{cl}_G(x^n)}n =\inf_{n\geq1}\frac{\operatorname{cl}_G(x^n)}n, \] where subadditivity ensures existence of the limit.

Prove or disprove that $\operatorname{scl}_G(x)\in\mathbb Q$ for every $x\in[G,G]$. The commutators in the minimum are taken in the whole surface group, even if a chosen word for $x$ lies in a smaller subgroup. Each finite ratio in the formula is rational, but its infimum need not be rational for that reason alone. The assertion requires exact rationality for every such element, rather than numerical approximations or bounds.

Short English statement

Is stable commutator length always rational for homologically trivial elements of a closed surface group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2779 (KP-2.31), 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 2.31. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.