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Recognition of positive one-relator groups

Develop an algorithm that decides whether a given finitely presented one-relator group admits a presentation with a relator containing no inverse letters (i.e., is a positive one-relator group).

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Background

Positive one-relator groups enjoy strong residual properties (e.g., residual solvability). Determining positivity is closely related to the isomorphism problem relative to this subclass and would enable algorithmic access to many structural consequences.

References

Note that e.g. the group $Z2 = #1{a,b}{[a,b] = 1}$ is not positive, by Proposition~\ref{Prop:Magnus-[a,b]-only-presentaton}. Positive one-relator groups enjoy some particular properties. ... the isomorphism problem of one-relator groups with respect to the class of positive one-relator groups is still an open problem, i.e. it is unknown whether we can decide if a given one-relator group is positive or not.

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 7.3 (Quotients and residual properties of one-relator groups)