Today we will see some methods of constructing groups.

**Definition.** Let be groups and let and be injective homomorphisms. If the diagram below is a pushout then we say write and we say that is the *amalgamated (free) product of and over .*

**Example.** If , we write and say is the *free product of and .*

As usual, we need to prove existence.

**Recall. **If is a group, then the *Eilenberg-MacLane Space* satisfies the following properties:

- is connected;
- ;
- for .

**Facts.**

- exists;
- The construction of is functorial;
- is unique, up to homotopy equivalence.

For as above, let and realize as a map and as a map . Now, let , where . By the Seifert-Van Kampen theorem, . Suppose that , and . Then,

.

In particular, if is finitely generated, then so is , and if are finitely presented and is finitely generated, then is finitely presented.

**Example.** Let be a connected surface and let be a separating, simple closed curve. Let . Then,

But, what if is non-separating (but still 2-sided)? Then, there are two natural maps representing , where . Associated to , we have a map , , which maps a curve to its signed (algebraic) intersection number with .

Let be a covering map corresponding to . Then,

This has a shift-automorphism . We can now recover :

**Defintion.** If are injective homomorphisms, then let

Let be the shift automorphism on . Now, is called the *HNN (*Higman, Neumann, Neumann) *Extension of over .* We often realize as , where and . It is easy to write down a presentation:. is called a *stable letter*.

## 5 comments

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5 March 2009 at 7.06 pm

pweilSo are HNN extensions anything special categorically? “almost coequalizers”?

5 March 2009 at 8.27 pm

Henry WiltonGood question – and I’m afraid I don’t know the answer. Perhaps someone from cyberspace can enlighten us.

6 April 2009 at 10.18 pm

katermanI think that the HNN extension is the coequalizer of the diagram where is conjugation by following . This seems to suggest that “” (i.e. ) is something special in …

15 October 2012 at 11.30 pm

Qiaochu YuanHNN extensions are pushouts in the category of groupoids. This was first pointed out to me by Ronnie Brown on math.SE. (The pushout diagram is a little annoying to and I’m not sure if anyone’s reading this. Exercise?)

16 October 2012 at 2.38 pm

Henry WiltonThanks for pointing this out, Qiaochu. My impression is that this is the major advantage of working with groupoids.