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Derived‑series stabilization conjecture for one‑relator groups

Establish that for every one‑relator group G = F/<<w>>, there exists r ∈ F with w ∈ r [<<r>>, <<r>>] such that the derived series stabilizes at G^{(ω)} = G^{(ω+1)} = <<r>>_G; equivalently, prove that the derived series length of G is at most ω and its maximal residually solvable quotient is F/<<r>>.

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Background

The authors conjecture a uniform stabilization of the derived series in one‑relator groups, linking solvable quotients to a single ‘residual relator’ r. This would mirror known stabilization phenomena in special families.

The conjecture arises from their structural results on rationally solvable quotients and would refine understanding of solvable approximations of one‑relator groups.

References

In it is conjectured that this is the case. \begin{conjecture} Let $G = F/{w}$ be a one-relator group. There is a word $r\in F$ such that $w\in r[{r}, {r}]$ and

G{(\omega+1)} = G{(\omega)} = {r}_G.

In particular, the maximal residually solvable quotient of $G$ is the one-relator group $F/{r}$. \end{conjecture}

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Section 7.1 (Residually solvable and rationally solvable one‑relator groups)