Finitely Presented Hopfian Group

• A finitely presented Hopfian group is defined by a finite set of generators and relators that guarantees every surjective endomorphism is an automorphism, ensuring algebraic rigidity.
• Its construction uses amalgamated products and hyperbolic 3-manifold groups to embed any countable group while preserving finiteness, torsion-freeness, and structural properties.
• Key properties such as completeness, trivial center, and the preservation of finite classifying spaces underscore its significance in geometric and combinatorial group theory.

A finitely presented Hopfian group is a group specified by a finite set of generators and relators, in which every surjective endomorphism is an automorphism. In the context of geometric and combinatorial group theory, these groups are central objects due to their algebraic rigidity and their embedding properties. Notably, recent work establishes that every countable group can be embedded into a finitely generated, finitely presented group that is simultaneously Hopfian and complete, where completeness entails having trivial center and every automorphism being inner (Bridson et al., 2023).

1. Definitions and Main Properties

A group $G$ is Hopfian if every surjective endomorphism $\phi:G \to G$ is necessarily an automorphism; equivalently, $G$ cannot be isomorphic to a proper quotient of itself. A finite presentation for $G$ means that $G$ can be written as

$$G = \langle x_1, \dots, x_n \mid r_1, \dots, r_m \rangle$$

for some finite sets of generators $\{x_i\}$ and relators $\{r_j\}$. The rigidity inherent to Hopfian groups is tied to group properties such as residual finiteness, co-Hopficity, and completeness. A group is complete if its center is trivial and every automorphism is inner. Finitely presented Hopfian groups with completeness and additional properties such as finite classifying spaces appear as key constructions in modern group theory.

2. Embedding Countable Groups into Finitely Presented Hopfian Groups

Every countable group $G$ can be embedded into a finitely generated group $G^*$ that is both Hopfian and complete. If $G$ is finitely presented (and, respectively, if $G$ admits a finite classifying space), then so is $G^*$. The embedding is constructed via a four-stage process involving group amalgams and the use of hyperbolic 3-manifold groups (Bridson et al., 2023). The construction preserves essential properties such as torsion-freeness and preservation of finite subgroups up to conjugacy.

3. Construction Methodology: Amalgamated Products and Hyperbolic 3-Manifolds

The construction begins by preparing $G$ so that it contains two free subgroups $F_1, F_2$ of specified rank with noncyclic intersection and trivial centralizer. Then, two asymmetric, co-Hopfian, non-Haken hyperbolic 3-manifold groups $A_1, A_2$ (each torsion-free, center-trivial, and possessing property FA) are produced, each containing malnormal free subgroups $L_1 \cong F_1$ and $L_2 \cong F_2$. The final group is obtained via the amalgamated free product: $G^* = A_1 *_{L_1=F_1} G *_{F_2=L_2} A_2$ with a combinatorially explicit, finite presentation formed by uniting the generators and relators of $G$, $A_1$, and $A_2$ and imposing appropriate identification relations.

The construction uses closed hyperbolic 3-manifolds arising from non-Haken knot complements and high-distance Dehn fillings (Thurston, Kojima, Hatcher) to ensure properties such as asymmetry ($\operatorname{Isom}(M) = \{1\}$), co-Hopficity, and malnormal free subgroups (Bridson et al., 2023).

4. Finiteness and Verification of Structural Properties

A detailed analysis of $G^*$ establishes the following:

• Trivial Center: In an amalgam where edge groups are non-central, the center of the amalgam is trivial. Here, each $L_i$ is free and center-trivial, so $Z(G^*) = \{1\}$.
• Completeness: Any automorphism of $G^*$ is induced by an inner automorphism. This is shown by analyzing the Bass–Serre tree associated to the amalgam and using the fact that $A_i$ are complete and co-Hopfian.
• Hopficity: If each $A_i$ has property FA—which is satisfied by non-Haken hyperbolic 3-manifold groups—then every surjective endomorphism of $G^*$ is an automorphism, following from injectivity on the $A_i$ and the action on the tree.
• Preservation of Finiteness: The amalgam remains finitely presented if its constituents are. If each factor admits a finite $K(-,1)$, so does the amalgam, which admits a finite 2-dimensional CW model for its classifying space.

5. Explicit Presentation and Summary Table

A finitely presented Hopfian group $G^*$ arising from this framework has the following presentation (with notation as above): $$G^* = \langle x_1, \dots, x_n, y^1_1, \dots, y^1_{d_1}, y^2_1, \dots, y^2_{d_2} \mid r_1, \dots, r_m, R^1_1, \dots, R^1_{s_1}, R^2_1, \dots, R^2_{s_2}, f^1_i(x) = l^1_i(y^1), f^2_j(x) = l^2_j(y^2) \rangle,$$ where matching of free subgroup generators and relators ensures finite presentability and the preservation of group-theoretic rigidity (Bridson et al., 2023).

Property	Description	Source
Hopfian	Every surjective endomorphism is an automorphism	(Bridson et al., 2023)
Complete	Trivial center and all automorphisms are inner	(Bridson et al., 2023)
Finitely presented	Presentation is finite if $G$ is finitely presented	(Bridson et al., 2023)
Finite classifying space	Finite $K(G^*,1)$ if $G$ (and $A_1$, $A_2$) have finite $K(-,1)$	(Bridson et al., 2023)
Torsion-free	Inherits torsion-freeness from the construction	(Bridson et al., 2023)

6. Role of Hyperbolic 3-Manifold Groups

The key technical ingredient is the existence of torsion-free, center-trivial, asymmetric, co-Hopfian hyperbolic 3-manifold groups with property FA and malnormal free subgroups. These arise as fundamental groups of closed hyperbolic 3-manifolds produced using high-distance Dehn fillings of non-Haken knot complements. Mostow–Prasad rigidity ensures their asymmetry, and their group-theoretic properties guarantee the rigidity required in the amalgam construction. Lemma 2.1 in (Bridson et al., 2023) provides explicit details for all ranks and dimensions $d \geq 3$.

7. Significance and Scope

This construction establishes that every countable group is not only embeddable in a finitely generated group but can be embedded into a group with maximal rigidity, i.e., both Hopfian and complete, while preserving finiteness properties of presentation and classifying space. Every finite subgroup of the constructed group is conjugate to a finite subgroup of the original, ensuring tight control over subgroup structure (Bridson et al., 2023). A plausible implication is increased flexibility for embedding problems while maintaining strong algebraic rigidity, informing both geometric and combinatorial group theory.