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

TL;DR

This paper constructs an explicit example of a non-Hopfian ascending HNN-extension of a finitely presented Hopfian group, addressing a question related to residual finiteness and Hopfian properties.

Contribution

It provides the first explicit construction of a non-Hopfian ascending HNN-extension of a finitely presented Hopfian group, answering an open question in group theory.

Findings

Constructed an explicit non-Hopfian ascending HNN-extension

Addresses an open question about residual finiteness and Hopfian groups

Links properties of finitely generated residually finite groups to Hopfian behavior

Abstract

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.