Submonoid membership in hyperbolic and torsion one-relator groups

Determine whether there exists (i) a hyperbolic one-relator group with undecidable submonoid membership problem, and (ii) a one-relator group with torsion with undecidable submonoid membership problem.

Background

Several one-relator groups are known to have undecidable submonoid membership via embeddings of right-angled Artin groups. However, these constructions do not apply to hyperbolic settings, leaving open whether hyperbolic or torsion one-relator groups can exhibit such undecidability.

References

Thus whether there is some hyperbolic one-relator group with undecidable submonoid membership problem, and whether there is a one-relator group with torsion with undecidable submonoid membership problem, both remain open problems, being Problems~20.68 {content} 20.69, respectively, in the Kourovka Notebook .

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Subsection 8.2 (More on the word and membership problems)