I'm due to start my (fully funded!) PhD in Mathematics this October (2017) and I'll be working closely with $H(r, n, s)$, so a detailed answer aimed at that level would be ideal.
The Details:
Definition 1: Let $F_n$ be the free group on $x_1, \dots, x_n$. Let $\sigma=(12\dots n)\in\mathcal S_n$. Let $\theta$ be the automorphism of $F_n$ which permutes $x_1, \dots, x_n$ by $\sigma$. If $w$ is a word in $F_n$, let $N_n(w)$ be the normal closure of $$\langle w, \theta(w), \dots, \theta^{n-1}(w)\rangle$$ and let $G_n(w)=F_n/N_n(w)$. Define $$H(r, n, s)=G_n(x_1x_2\dots x_r(x_{r+1}\dots x_{r+s})^{-1}).$$
Definition 2: Define $$I(r, n, s)=\langle x, t\mid x^st^r=t^sx^r, t^n=1\rangle.$$
The Question:
How is $I(r, n, s)$ a semi-direct product of $H(r, n, s)$ with $C_n$?
Thoughts:
I don't have much to add, I'm afraid. I've looked to see if there are any nice presentations for the semi-direct product of two groups but to no avail.
Here $H(r, n, s)$ has the obvious presentation defined by the factor group $F_n/N_n(w)$.
(NB: I've found it! I know how it's done. I'll share it in an answer once I have time.)
Please help :)
