I finished an MPhil a year ago that focused on the following question. I've moved on to a different area of group theory now, so I thought I'd ask it here.
Definition: Let $w\in F_n$ for the free group $F_n$ on $\{x_i\mid i\in\overline{0,n-1}\}$ and $n\in \Bbb N$. A cyclically presented group is a group $G_n(w)$ with presentation
$$P_n(w):=\langle x_0, \dots, x_{n-1}\mid w, \theta(w), \dots, \theta^{n-1}(w)\rangle,$$
where $$\begin{align}\theta: F_n&\to F_n,\\ x_i&\mapsto x_{i+1\pmod{n}}.\end{align}$$
Let $w=x_0x_jx_kx_l$. This is a positive word of length four.
Definition: A class $\mathcal{G}$ of groups satisfies the Tits Alternative if for any $G$ in $\mathcal G$ either $G$ has a free, non-abelian subgroup or $G$ has a solvable subgroup of finite index.
The Question:
Do cyclically presented groups of positive word length four relators satisfy the Tits Alternative?
My dissertation is available here and here. A paper by myself and my supervisor at the time, Prof. G. Williams, on the topic can be found here and here.
We found that, using mainly Bogley & Parker's paper [1], for
$$ (A)=\left\{\begin{array}{rlc} 2k &\equiv 0& \text{or}\\ 2j &\equiv 2l& \end{array}\right. $$
$$(B)=\left\{\begin{array}{rlc} k &\equiv 2j& \text{or}\\ k &\equiv 2l& \text{or}\\ 2k&\equiv j+l& \text{or}\\ 0&\equiv j+l& \end{array}\right.$$
$$(C)=\left\{\begin{array}{rlc} l&\equiv j+k& \text{or}\\ j &\equiv l+k& \end{array}\right.$$
all modulo $n$, in all cases but when $(A)$ and $(C)$ are false while $(B)$ is true, the Tits Alternative holds. We gathered evidence in GAP for the remaining case up to $n=20$.
References:
[1] Bogley, W., A., Parker, F., W. (2018) Cyclically Presented Groups with Length Four Positive Relators. J. Group Theory 21(5) (2018).
