---
title: Non-Hopfian Ascending HNN-Extension
url: https://www.emergentmind.com/topics/non-hopfian-ascending-hnn-extension
type: topic
---

# Non-Hopfian Ascending HNN-Extension

A non-Hopfian ascending HNN-extension is a group constructed as an HNN-extension of a base group \(G\) along an injective but non-surjective endomorphism, resulting in a group \(H\) that admits a surjective, non-injective endomorphism—that is, \(H\) is not Hopfian, even when \(G\) is. Recent work has produced explicit examples where the base is finitely presented and Hopfian, settling a question regarding the analog of Sapir–Wise’s non-residually finite constructions in the context of the Hopf property [2512.09135].

## 1. Ascending HNN-Extensions and the Hopf Property

Consider a group \(G\) and an injective homomorphism \(\phi: G \to G\). The ascending HNN-extension associated to \(\phi\) is the group
\[
H = \left\langle G,\,t \mid t^{-1}gt = \phi(g) \text{ for every } g \in G \right\rangle.
\]
A group is Hopfian if every surjective endomorphism is an automorphism. An ascending HNN-extension can fail to be Hopfian if there exists a surjective, non-injective self-map, often constructed by lifting a suitable endomorphism \(\psi: G \to G\) to \(H\) so that the extension \(\widetilde{\psi}: H \to H\), defined on \(G\) as \(\psi\) and fixing \(t\), is surjective but not injective. The presence of an injective but non-surjective \(\phi\) and a commuting, non-injective \(\psi\) is the essential mechanism for this failure.

## 2. Explicit Construction of a Non-Hopfian Ascending HNN-Extension

The construction in [2512.09135] begins with the group
\[
G = \langle a, b, s \mid b^{-1} a b = a,\;\; s^{-1} a^2 s = a^4 \rangle.
\]
This group can be alternatively described as an HNN-extension over \(\mathbb{Z}^2\), using the stable letter \(s\) to identify the subgroup \(\langle a^2 \rangle\) with \(\langle a^4 \rangle\). The presentation is inspired by the Andreadakis–Raptis–Varsos theorem that groups of the form
\[
\langle a, b, s \mid b^{-1}ab = a,\;\; s^{-1}a^p s = a^q \rangle
\]
are Hopfian whenever \(1 < p < q\). The injective endomorphism \(\phi: G \to G\) is given by
\[
\phi(a) = a^2,\quad \phi(b) = b, \quad \phi(s) = s^{-1} b s^2.
\]
This map preserves all relators, is injective but not surjective (\(a \not\in \phi(G)\)), and allows the formation of the ascending HNN-extension
\[
H = \langle a, b, s, t \mid b^{-1}ab = a,\, s^{-1}a^2s = a^4,\, t^{-1}a t = a^2,\, t^{-1}b t = b,\, t^{-1} s t = s^{-1} b s^2 \rangle.
\]
The base \(G\) is verified to be Hopfian, but not co-Hopfian, by invoking the cited results.

## 3. Surjective Non-Injective Endomorphism and Non-Hopficity

To show \(H\) is not Hopfian, the construction employs Sapir–Wise’s Lemma: If in addition to \(\phi\) there is an endomorphism \(\psi: G \to G\) with
\[
\psi(a) = a^2,\quad \psi(b) = b, \quad \psi(s) = s,
\]
such that \(\psi\) and \(\phi\) commute, \(\phi(G) \subset \psi(G)\), and \(\psi\) is non-injective but still maps relators to the identity, then the extension \(\widetilde{\psi}: H \to H\) with \(\widetilde{\psi}|_G = \psi\), \(\widetilde{\psi}(t) = t\), is a surjective, non-injective endomorphism of \(H\). In this case, the element \(s^{-1} a s a^{-2} \neq 1\) in \(G\) is mapped to the identity, proving non-injectivity. Hence, despite \(G\) being Hopfian, the group \(H\) constructed by adjoining \(t\) according to the endomorphism \(\phi\) is non-Hopfian [2512.09135].

## 4. Contrasts with Free Group Ascending HNN-Extensions

For ascending HNN-extensions of free groups, the global criterion for Hopficity is determined by the presence of Baumslag–Solitar (\(\mathrm{BS}(1, n)\)) subgroups. Specifically, the group
\[
G = \langle F, t \mid t^{-1} x t = \phi(x) \text{ for all } x \in F \rangle
\]
with \(F\) free and \(\phi\) injective is Hopfian if and only if it contains no subgroup isomorphic to any \(\mathrm{BS}(1, n)\) (\(n \geq 2\)). If such a subgroup exists, the HNN-extension is necessarily non-Hopfian [2005.11896]. In the free group context, hyperbolicity exactly coincides with Hopficity and the absence of Baumslag–Solitar subgroups:
\[
\text{Hopfian} \iff \text{hyperbolic} \iff \text{no } \mathrm{BS}(1, n).
\]
The explicit construction in [2512.09135], by contrast, uses a base group that is not free, and whose structure and endomorphisms enable the Hopficity failure independently of Baumslag–Solitar obstructions.

## 5. Relation to Mal’cev’s Theorem, Residual Finiteness, and Further Directions

Mal’cev’s theorem asserts that every finitely generated residually finite group is Hopfian. Sapir and Wise constructed non-residually finite ascending HNN-extensions of finitely generated residually finite groups, showing that the extension can be non-Hopfian even when the base is residually finite. The parallel question—whether there exists a non-residually finite ascending HNN-extension of a finitely presented residually finite group—remains open. However, the result in [2512.09135] provides the Hopfian analogue: an explicit non-Hopfian ascending HNN-extension of a finitely presented Hopfian group. This advances understanding of the interplay between residual finiteness, presentation finiteness, Hopficity, and mapping-torus constructions.

Additionally, Sapir–Wise conjectured that if the base group is hyperbolic, then the ascending HNN-extension remains Hopfian. The construction of [2512.09135] suggests that finding a non-Hopfian example in the hyperbolic case would require a Hopfian, non-co-Hopfian hyperbolic group with the requisite commuting endomorphisms. Known non-Hopfian ascending HNN-extensions, including these constructions, are not relatively hyperbolic, maintaining a boundary with Gromov’s residual finiteness questions.

## 6. Comparison of Base Group Properties and Significance

The construction in [2512.09135] achieves the smallest known example of a mapping torus with a “Hopfian but not co-Hopfian” base:
\[
G = \langle a, b, s \mid b^{-1}ab = a,\, s^{-1}a^2s = a^4 \rangle.
\]
The Hopficity of \(G\) (guaranteed by Andreadakis–Raptis–Varsos for such presentations with \(2 < 4\)) contrasts with the existence of an injective, non-surjective \(\phi\), reflecting the delicate distinction between Hopficity and co-Hopficity. The ability to construct a non-Hopfian ascending HNN-extension above such a base underscores the subtlety of endomorphism dynamics in group extensions and the limitations of extending Hopficity to mapping tori even for well-behaved base groups [2512.09135].

Source: https://www.emergentmind.com/topics/non-hopfian-ascending-hnn-extension