I suppose I've answered my own question in writing it but some input would be great.
The Question(s)
What exactly are cyclically presented groups, what are some examples, and where might I find out more about them?
What I have so far.
Definition 1: (This was found here.) Cyclically presented groups, such as the Fibonacci groups, are ones defined by a presentation with a cyclic symmetry.
This is okay, I guess, but I'm still not a hundred percent sure on the definition; what is meant by a "cyclic symmetry"? I suppose my next definition, which I found in a paper by Prof. G. Williams, explains it.
Definition 2: Let $w=w(x_0, \dots , x_{n-1})$ be a word in the free group $F_n$ with generators $x_1, \dots , x_n$ and let $\theta$ be the automorphism given by
$$\begin{align} \theta:& F_n \to F_n \\ \, & x_i \mapsto x_{i+1}. \end{align}$$
A cyclically presented group is a group $G_n(w)$ with the cyclic presentation $$\mathcal G_n(w)=\langle x_0, \dots , x_{n-1}\mid w, \theta(w), \dots , \theta^{n-1}(w)\rangle.$$
As for examples, we have the Fibonacci groups given by the presentation $$\langle x_0, \dots , x_{n-1}\mid x_ix_{i+1}=x_{i+2}\rangle,$$ which include the Quaternion group $Q_8$ and the cyclic groups $\Bbb Z_5, \Bbb Z_{11}$, and $\Bbb Z_{29}$. They're $G_n(x_0x_1x_2^{-1})$.
We also have groups of Fibonacci type, $G(x_0x_kx_m^{-1})$, which include $SL(2,3)$, $SL(2, 5)$, and $\Bbb Z_7$.
As to where to find out more, there's the last chapter of "Presentations of Groups," by D. L. Johnson. This would be great but it's at the end of a book, so might not be accessible (at least for a while, like, say, a couple of months).
Please help :)
