Foliations and Diffeomorphism Groups — Question 51

Definitions and mathematical statement

For a fixed integer $g\geq 1$, let $\Sigma_g$ denote a surface of genus $g$. For a precise conditional formulation, interpret $\Sigma_g$ as a closed, connected, oriented smooth surface. These surface hypotheses are conventional but are not explicitly stated in the supplied record and require confirmation.

Write \[ D_g=\mathrm{Diff}_0^\infty(\Sigma_g),\qquad H_g=\mathrm{Homeo}_0(\Sigma_g), \] where the subscript $0$ denotes the identity component, using the $C^\infty$ topology for diffeomorphisms and the compact-open topology for homeomorphisms. These topology conventions also require confirmation against the source context.

For a group $G$, define $[a,b]=aba^{-1}b^{-1}$, and let $[G,G]$ be the subgroup generated by commutators. For $f\in[G,G]$, its commutator length and stable commutator length are \[ \mathrm{cl}_G(f)=\min\left\{k\in\mathbb Z_{\geq 0}: f=\prod_{i=1}^{k}[a_i,b_i],\ a_i,b_i\in G\right\}, \qquad \mathrm{scl}_G(f)=\lim_{n\to\infty}\frac{\mathrm{cl}_G(f^n)}{n}. \] The empty product is the identity. All commutators and lengths below are taken in the indicated ambient group.

The two questions, with $g$ treated as a fixed arbitrary genus, are whether \[ \exists f\in[D_g,D_g]\quad \left(\mathrm{scl}_{D_g}(f)=0\ \text{and}\ \forall a,b\in D_g,\ f\neq[a,b]\right), \] and whether \[ \exists f\in[H_g,H_g]\quad \left(\mathrm{scl}_{H_g}(f)=0\ \text{and}\ \forall a,b\in H_g,\ f\neq[a,b]\right). \] Equivalently, in each question one asks for an element with stable commutator length zero and commutator length at least two. Membership in the commutator subgroup specifies the domain of the stable commutator length convention used here. The supplied wording does not explicitly distinguish existence for every genus from existence for at least one genus; answers should therefore specify their genus range.

As background, the supplied record attributes unboundedness of stable commutator length on the diffeomorphism groups under discussion to work of Bowden, Hensel and Webb using their fine curve graph, with reference label bowden2022quasi. This is a supplied contextual claim, not an independently verified theorem assessment. The opening sentence also mentions asking "the same question" in $\mathrm{Diff}_0^\infty(S^2)$, but its antecedent is absent. A precise sphere question cannot be recovered from this record alone.

Short English statement

For a surface of fixed genus $g\geq1$, ask whether its identity component of smooth diffeomorphisms contains an element whose stable commutator length is zero but which is not a single commutator. Ask the analogous question for the identity component of homeomorphisms. The closed, oriented surface convention, the intended genus quantifier, and the incomplete sphere reference require source confirmation.

Sources

• Local UnsolvedMath registry: permanent ID $800110$, exact source problem code catalogue entry, title "Foliations and Diffeomorphism Groups --- Question 51." The supplied registry labels the problem open; that label is supplied metadata, not an independently verified status assessment. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Nariman and Yazdi, Problem List on Foliations and Diffeomorphism Groups (2025), Question 51, https://arxiv.org/abs/2503.18274. The record attributes the problem to Nariman and Yazdi and records extraction from arXiv LaTeX, with access date 2026-10-16. These provenance details are supplied; no source visit or independent verification is claimed.
• Bowden, Hensel and Webb, cited in the supplied statement under reference label bowden2022quasi, for work using their fine curve graph and the contextual assertion of unbounded stable commutator length. No full bibliographic entry or separate URL for this reference is supplied.