A non-Hopfian ascending HNN-extension of a finitely presented Hopfian group

We find a non-Hopfian ascending HNN-extension of a finitely presented Hopfian group by providing an explicit construction. This result addresses an analogous question to the one posed by Sapir and Wise, which asks whether there is a non-residually finite ascending HNN-extension of a finitely presented residually finite group. Such an analogy is motivated by Mal’cev’s result that every finitely generated residually finite group is Hopfian.

💡 Research Summary

The paper addresses a natural analogue of a question posed by Sapir and Wise concerning ascending HNN‑extensions. While Sapir‑Wise exhibited a non‑residually‑finite ascending HNN‑extension of a finitely generated residually finite group (the base group being infinitely presented), the present work asks whether one can obtain a non‑Hopfian ascending HNN‑extension when the base group is both finitely presented and Hopfian. The authors answer this in the affirmative by constructing an explicit example.

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