I present here a series of four notes about constructions using amalgamated products and HNN extensions. These operations are central in geometric group theory, thanks to van Kampen theorem and Bass-Serre theory (both are in fact related, see Scott and Wall’s article, Topological method in group theory). We begin by defining them:

Definition: Let $A,B,C$ be three groups and $\varphi : C \to A$, $\psi : C \to B$ be two monomorphisms. If $\langle X \mid R \rangle$ (resp. $Y \mid S \rangle$) is a presentation of $A$ (resp. of $B$), we define the amalgamated product $A \underset{C}{\ast} B$ by the presentation

$\langle X, Y \mid R, S, \varphi(c)= \psi(c) \ \forall c \in C \rangle$.

Therefore, we glue $A$ and $B$ together along $C$.

Definition: Let $A, C$ be two groups and $\varphi : C \to A$, $\psi : C \to A$ be two monomorphisms. If $\langle X \mid R \rangle$ is a presentation of $A$, we define the HNN extension $\underset{C}{\ast} A$ by the presentation

$\langle X, t \mid R, t \varphi(c) t^{-1} = \psi(c) \ \forall c \in C \rangle$.

We say that $t$ is the stable letter of the extension.

Therefore, we force the two isomorphic subgroups $\varphi(C)$ and $\psi(C)$ to be conjugated.

It is worth noticing that the amalgamated products and the HNN extensions depend on the monomorphisms we chose, although they do not appear in the notations. However, the new groups we build do not depend on the specific presentations we use; it can be noticed directly. Alternatively, these operations may be defined thanks to universal properties, whithout refering to any presentation.

A remarkable fact is that normal forms are known for amalgamated products and HNN extensions:

Property: Let $A \underset{C}{\ast} B$ be an amalgamated product. Let $T_A$ (resp. $T_B$) be a transversal of $A$ (resp. $B$) modulo $C$; for the coset $C$, we take $1$ as a coset representative. Any element of $A \underset{C}{\ast} B$ may be uniquely written as

$c \cdot a_1 \cdot b_1 \cdots a_n \cdot b_n$,

where $n \geq 1$, $a_i \in T_A \backslash \{1\}$ for $2 \leq i \leq n$, $a_1 \in T_A$, $b_i \in T_B \backslash \{1\}$ for $1 \leq i \leq n-1$, $b_n \in T_B$ and $c \in C$.

Property: (Britton’s lemma) Let $\underset{C}{\ast} A$ be a HNN extension. Let $T_1, T_2$ be transversals of the images of $C$ into $A$; for the trivial cosets, we take $1$ as a coset representative. Any element of $\underset{C}{\ast} A$ may be uniquely written as

$a_1 \cdot t^{\epsilon_1} \cdots a_n \cdot t^{\epsilon_n} \cdot a_{n+1}$,

where $n \geq 1$, $a_{n+1} \in A$, $a_i \in T_1$ if $\epsilon_i = 1$, $a_i \in T_2$ if $\epsilon_i = -1$, and $a_i \neq 1$ if $\epsilon_i \neq \epsilon_{i+1}$.

In particular, it is possible to deduce the following corollaries, which will be useful in the sequel:

Corollary: The factors $A,B$ of an amalgamated product $A \underset{C}{\ast} B$ embed naturally into $A \underset{C}{\ast} B$. The factor $A$ of a HNN extension $\underset{C}{\ast} A$ embeds naturally into $\underset{C}{\ast} A$.

Corollary: Up to conjugaison, an element of finite order in an amalgamated product of a HNN extension belongs to a factor. In particular, the amalgamated product of torsion-free groups or the HNN extension of a torsion-free group is torsion-free.

Our first note deals with an embedding theorem:

Theorem: Every countable group can be embedded into a two-generator groups.

(Notice that we gave a proof of this theorem without using amalgamated products and HNN extensions in the note A free group contains a free group of any rank.)

As a corollary, we prove the following theorem of B. H. Neumann that we already proved using the space of marked groups and small cancellation groups in the note Cantor-Bendixson rank in group theory: A theorem of B.H. Neumann:

Theorem: Up to isomorphism, there exist $2^{\aleph_0}$ two-generator groups.

A consequence of the theorem above is that almost all finitely-generated groups are not finitely presented: clearly, there exist only countably many finitely-presented groups. However, in general it is not easy to prove that a group is not finitely-presented of even to find such a finitely-generated group. It turns out that amalgamated products and HNN extensions can be used to give examples of finitely-generated not finitely-presented groups. More precisely, we prove

Theorem: Let $A,B$ be two finitely-presented groups. The amalgamated product $A \underset{C}{\ast} B$ is finitely-presented if and only if $C$ is finitely-generated.

Theorem: Let $A$ be a finitely-presented group. The HNN extension $\underset{C}{\ast} A$ is finitely-presented if and only if $C$ is finitely-generated.

We also use Britton’s lemma to deal with some lamplighter groups:

Theorem: The groups $\mathbb{Z}_n \wr \mathbb{Z}$ and $\mathbb{Z} \wr \mathbb{Z}$ are not finitely-presented.

Another kind of counterexamples provided by amalgamated products and HNN extensions is the class of finitely-generated non-Hopfian groups.

A group $G$ is said Hopfian if every epimorphism $G \twoheadrightarrow G$ is in fact an isomorphism. Several years was needed to fin a non-Hopfian finitely-generated group, and the first examples which were given are precisely based on amalgamated products and HNN extensions. Defining the Baumslag-Solitar group $BS(n,m)$ by the presentation

$\langle a,t \mid ta^nt^{-1}=a^m \rangle$,

we prove:

Theorem: Let $n,m \in \mathbb{Z} \backslash \{0\}$ be two coprime numbers with $n \neq \pm 1$. Then $BS(n,m)$ is not Hopfian.

Another embedding theorem that can be proved using amalgamated products and HNN extensions is the following one:

Higmann embedding theorem: A group can be embedded into a finitely-presented group if and only if it is recursively-presented.

As a corollary, it may be deduced that there exists a finitely-presented group with an insoluble word problem. Unfortunately, the proof of this result is much more evolved that the previous ones, and I had not the time to write such a proof. By the way, I hope to be able to add a note on this theorem soon; meanwhile, a proof can be found in Lyndon and Schupp’s book.

If we suppose that there exists a finitely-presented group with an insoluble word problem, it turns out that many other decision problems can be shown to be insoluble. More precisely, if we define a Markov property $\mathcal{M}$ as a property of finitely-presented groups, preserved by isomorphism, and such that there exist a finitely-presented group satisfying $\mathcal{M}$ and a finitely-presented group not embeddable into any finitely-presented group satisfying $\mathcal{M}$, we prove:

Theorem: Let $\mathcal{M}$ be a Markov property. There is no algorithm to decide whether or not a finitely-presented group satisfies $\mathcal{M}$.

In particular,

Corollary: There is no algorithm to decide whether or not a finitely presented group is abelian, finite, trivial, free, torsion-free or cyclic.

Corollary: There is no algorithm to decide whether or not two presentations represent the same group.

Because any finitely-presented group is the fundamental group of a closed $n$-manifold for any $n \geq 4$, several facts about decision problems in topology may be deduced. For example,

Corollary: Let $n \geq 4$. There is no algorithm to decide whether two closed $n$-manifolds are homeomorphic (or even simply connected).

The main references I used in this series of notes are Lyndon and Schupp’s book, Combinatorial Group Theory, and Baumslag’s book, Topics in Combinatorial Group Theory.