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Undistortion conjecture for cubulated one‑relator groups

Characterize undistortion of finitely generated subgroups of cubulated one‑relator groups by proving that every finitely generated subgroup is undistorted if and only if π(w) ≠ 2 or π(w) = 2 and the w‑subgroup P ≤ G is virtually F_n × Z for some n ≥ 1.

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Background

Cubulated one‑relator groups (virtually special) have powerful geometric and separability properties, but subgroup distortion can occur (e.g., hydra groups). The conjecture proposes a precise criterion in terms of primitivity rank and the structure of the w‑subgroup.

It would connect algebraic primitives to geometric undistortion in virtual special settings.

References

We conclude with a conjecture. If $G$ is a cubulated one-relator group, then every finitely generated subgroup of $G$ is undistorted if and only if $\pi(w) \neq 2$ or $\pi(w) = 2$ and $P \leqslant G$ is virtually $F_n\times Z$ for some $n\geqslant 1$.

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Section 6.3 (Virtually compact special one‑relator groups)