Skip to main content

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.

So how about this as a presentation for an FPA?

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

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?