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Chong–Wise embedding conjecture for free‑by‑cyclic groups

Prove that every finitely generated free‑by‑cyclic group embeds as a subgroup of a free‑by‑cyclic group whose free kernel is finitely generated.

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Background

Ascending HNN‑extensions of free groups admit embeddings into finitely generated versions; Chong–Wise conjecture analogous embeddings for general free‑by‑cyclic groups. Such embeddings would upgrade virtual fibring results from free‑by‑cyclic to {finitely generated free}‑by‑cyclic settings.

This has implications for residual finiteness, specialness, and algorithmic properties.

References

Conjecture [Chong--Wise] If $G$ is a finitely generated free-by-cyclic group, then $G$ is a subgroup of a {finitely generated free}-by-cyclic group.

— The theory of one-relator groups: history and recent progress (2501.18306 - Linton et al., 30 Jan 2025) in Section 4.2 (Virtual algebraic fibring and generalisations)