Kourovka Notebook Problem 5.42

Definitions and mathematical statement

Let $F=F(x,y)$ be the free group of rank $2$. For $w\in F$, define the verbal subgroup generated by $w$ in $F$ by \[ V_w(F)=\langle w(a,b):a,b\in F\rangle =\langle \varphi(w):\varphi\in\operatorname{End}(F)\rangle, \] where $w(a,b)$ denotes substitution of $a,b$ for $x,y$, and $\operatorname{End}(F)$ is the set of group endomorphisms of $F$. Thus generation by a single element here means generation as a verbal subgroup, rather than generation as an ordinary cyclic subgroup.

Does there exist a sequence $(w_n)_{n\geq 0}$ of elements of $F$ such that \[ V_{w_n}(F)\subsetneq V_{w_{n+1}}(F) \qquad\text{for every integer }n\geq 0? \] Here an infinite ascending chain is interpreted as a strictly ascending chain, as is customary in this question. The supplied registry labels the problem open; that label is supplied source metadata, not an independently verified assessment.

Short English statement

Can the free group on two generators contain an infinite strictly ascending chain of verbal subgroups, each generated as a verbal subgroup by one word?

Sources

• Local UnsolvedMath registry, permanent ID $700324$, exact source code catalogue entry, title "Kourovka Notebook Problem 5.42." The supplied record labels the problem open.
• Supplied source attribution: The Kourovka Notebook (Unsolved Problems in Group Theory), 5th issue (1976), Problem 5.42; also identified with the 21st issue edition (2026), arXiv:1401.0300v48. These references are supplied by the record and have not been independently checked.
• The supplied collection description names E. I. Khukhro and V. D. Mazurov as editors. The proposer field is malformed: its literal content is A. Yu. Ol'shanski\textbackslash u{\textbackslash i. The background repeats this attribution, and the supplied proposal year is 1976. The intended complete spelling is unresolved.