Kourovka Notebook Problem 14.14

Definitions and mathematical statement

Let $F_2=\langle a,b\rangle$ be the free group of rank two. For $x,y\in F_2$, put $[x,y]=x^{-1}y^{-1}xy$, and let $F_2'=[F_2,F_2]$ be its derived subgroup. For $g\in F_2'$, define its commutator length by \[ \operatorname{cl}_{F_2}(g)=\min\left\{r\in{\mathbb{Z}}_{\geq 0}:g=\prod_{i=1}^{r}[x_i,y_i],\quad x_i,y_i\in F_2\right\}. \] The empty product is the identity, so $\operatorname{cl}_{F_2}(1)=0$.

Let $\mathcal N$ denote the natural numbers in the source's convention. The supplied record does not specify whether $0\in\mathcal N$; this boundary convention remains unresolved. A pair $(k,m)\in\mathcal N^2$ is called admissible if \[ \exists w\in F_2'\qquad \operatorname{cl}_{F_2}(w^m)=k. \] The problem is to determine exactly the set \[ \left\{(k,m)\in\mathcal N^2:\exists w\in F_2',\ \operatorname{cl}_{F_2}(w^m)=k\right\}. \] A complete formulation must specify whether $\mathcal N={\mathbb{Z}}_{\geq1}$ or $\mathcal N={\mathbb{Z}}_{\geq0}$.

The supplied source reports the following known partial result, attributed to V. G. Bardakov: for every integer $k\geq2$, the pair $(k,2)$ is admissible. This is retained as a source-reported result, without independent verification.

Short English statement

Determine which pairs of natural numbers $(k,m)$ occur as the commutator length $k$ of an $m$-th power of an element of the derived subgroup of the free group of rank two. The source leaves the inclusion of zero among natural numbers unspecified and reports that every $(k,2)$ with $k\geq2$ is admissible.

Sources

• Local UnsolvedMath registry, permanent ID $700999$, exact source code catalogue entry, title Kourovka Notebook Problem 14.14; supplied upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82. The registry labels the problem open; this is a supplied classification, not an independently verified assessment.
• The Kourovka Notebook (Unsolved Problems in Group Theory), original problem 14.14, 14th issue (1999); the supplied background identifies the consulted edition as the 21st issue edition (2026), arXiv:1401.0300v48. The collection description names E. I. Khukhro and V. D. Mazurov as editors. These bibliographic details are supplied and have not been independently checked.
• Attribution: the statement begins with the names C. C. Edmunds and G. Rosenberger; the background and proposer field instead identify V. G. Bardakov, with proposed year 1999. The supplied record does not explain these differing attribution roles.
• V. G. Bardakov, Algebra and Logic, 39 (2000), 224--251, cited in the supplied statement for admissibility of every pair $(k,2)$ with $k\geq2$. No article title or external link is supplied.