Decidability of residual finiteness for one‑relator groups

Determine whether residual finiteness is a decidable property in the class of one‑relator groups; that is, develop an algorithm that, given a one‑relator presentation F(S)/<<w>>, decides whether the group is residually finite.

Background

Residual finiteness is a weaker property than free‑by‑cyclic; numerous subclasses of one‑relator groups are known to be residually finite, and many (e.g., Baumslag–Solitar groups) are not. The authors point out that the existence of a deciding algorithm across the full class remains unsettled.

Progress on this would bridge algorithmic group theory and profinite methods and impact the classification of one‑relator groups.