Submonoid membership problems in hyperbolic and torsion one‑relator groups

Determine whether there exists a hyperbolic one‑relator group with undecidable submonoid membership problem and whether there exists a one‑relator group with torsion with undecidable submonoid membership problem.

Background

Submonoid membership generalizes subgroup membership and is known to be undecidable in several right‑angled Artin groups, which embed in many one‑relator groups. The authors note that all current undecidability examples factor through non‑hyperbolic embeddings, leaving the hyperbolic and torsion one‑relator cases unresolved.

These specific open instances are codified as Problems 20.68 and 20.69 in the Kourovka Notebook.