Non-Hopfian Ascending HNN-Extension

• Non-Hopfian ascending HNN-extension is a group constructed via an injective but non-surjective endomorphism that produces a surjective, non-injective mapping.
• It employs a finitely presented Hopfian base group with carefully designed endomorphisms to challenge traditional Hopficity in group extensions.
• The construction connects to debates on residual finiteness and hyperbolicity, offering insights into mapping-torus behavior and extension dynamics.

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 (Kim et al., 9 Dec 2025).

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 (Kim et al., 9 Dec 2025) 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 (Kim et al., 9 Dec 2025).

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 (Mutanguha, 2020). 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 (Kim et al., 9 Dec 2025), 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 (Kim et al., 9 Dec 2025) 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 (Kim et al., 9 Dec 2025) 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 (Kim et al., 9 Dec 2025) 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 (Kim et al., 9 Dec 2025).