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 .
- 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.