# Some notation for FPAs and HNN extensions

In order to define a free product with amalgamation $$A \ast_C B$$ you need to define $$A,B,C$$ and embeddings of $$C$$ into $$A$$ and $$B$$. Alternatively, instead of $$C$$ you can use a homomorphism $$phi: A \rightarrow B$$ and its inverse. A third option is to pick subgroups $$D \subset A$$ and $$E \subset B$$ and give an isomorphism between them.

All three of these options require you to define some groups and some morphisms. You can define a group nicely by giving a presentation but defining maps usually takes a few lines. So, I’d like a quicker way of describing the kinds of maps we need here.

Let’s take the third option. If we have generating sets so that $$D = \langle X \rangle$$ and $$E = \langle Y \rangle$$, then if we can write each of the elements of $$X$$ as words in $$Y^*$$ then that defines the required isomorphism completely. After you’ve done that, the generating set $$Y$$ might as well just be those words.

$\langle a_1, a_2, \dots | R_A \rangle [ x_1 = y_1, x_2 = y_2, \dots ] \langle b_1, b_2, \dots | R_B \rangle$

The bits in angle brackets are presentations of $$A$$ and $$B$$. The $$x_i$$ are words from $$A$$ and generate $$D$$, and the $$y_i$$ are words from $$B$$ and generate $$E$$.

Because this is all on one line and delimited adequately, you can chain things together and use brackets and all that, like this contrived example:

$\left ( \langle a,b \rangle [ b = c ] \langle c,d | cd = dc \rangle \right ) [ a=e ] \langle e,f \rangle$

For HNN extensions, you just need to say what the original group is, what the extending letter is, and what the identified subgroups are. So how about something like:

$\langle g_1, g_2, \dots | R_G \rangle [ t | x_1 = y_1, x_2 = y_2, \dots ]$

Apart from the different kinds of brackets being hard to distinguish visually, what have I missed?