This is the third note of our series dedicated to constructions using amalgamated products and HNN extensions.
In the early 1940’s, Hopf asked whether it is possible for a finitely-generated free group to be isomorphic to one of its proper quotients. Later, he proved that it is in fact impossible, but he left open the same question for finitely-generated groups. If a group is said Hopfian precisely when it is not isomorphic to one of its proper quotients, or equivalently, that any epimorphism turns out to be an isomorphism, the question becomes : Does there exist a non-Hopfian finitely generated group?
The first example of non-Hopfian finitely-generated groups was only found in 1950 by B. H. Neumann, see his article A two-generator group isomorphic to a proper factor group. In 1951, Higman gave the first example of non-Hopfian finitely presented group using amalgamated products:
Theorem: (Higman) The group is not Hopfian.
Proof. Let denote the additive group of dyadic rationals. We first prove that
is a presentation of . Clearly, the map sending to for all extends to an epimorphism . It is sufficient to prove that is one-to-one to conclude. Let
In , notice that for all ,
Therefore, if ,
But , hence . However, is an abelian group whose generators are sending to infinite-order element in by , so is torsion-free; in particular, we necessarily have . We just found our presentation.
Afterwards, let denote the semi-direct product where acts by multiplication by . Then
The third relation implies , so the presentation above may be simplified as
Let and be two copies of . From the discution above, we know that and have an infinite order in and respectively (in fact, recognizing a HNN extension, it immediatly follows from Britton’s lemma). Therefore, we may define the amalgamated product
Let (resp. ) be a morphism sending to and to (resp. to and to ). Noticing that and agree on , we may define .
We first notice that is onto. Indeed, , , and .
Then, we notice that is not one-to-one. Let . In ,
so . On the other hand,
We justify that (and in the same way that ) by saying that otherwise there would be a such that , hence
a contradiction with the fact that is torsion-free.
In 1962, Baumslag and Solitar found a simple family of non-Hopfian one-relator groups using HNN extensions.
Definition: Let . The Baumslag-Solitar group is the HNN extension of with respect to the subgroups and , and the obvious isomorphisms.
In particular, is the semi-direct product we described above.
Theorem: If are coprime with , then is not Hopfian.
Proof. Let be the morphism sending to and to ; is well-defined since
First, is onto. Indeed, and
where are two integers such that (they exist because and are coprime).
Then, we notice that is not one-to-one. Because , we deduce from Britton’s lemma that
On the other hand,