Non-freeness of groups generated by two parabolic elements with small rational parameters

Let $q\in \mathbb{C}$, let $$a=\left(\begin{array}{cc}1& 0\\ 1& 1\end{array}\right),b_q=\left(\begin{array}{cc}1& q\\ 0& 1\end{array}\right),$$ and let $G_q<{\mathrm{S}\mathrm{L}}_2(\mathbb{C})$ be the group generated by $a$ and $b_q$. In this paper, we study the problem of determining when the group $G_q$ is not free for $\mathrm{\mid }q\mathrm{\mid }<4$ rational. We give a robust computational criterion which allows us to prove that if $q=s\mathrm{/}r$ for $\mathrm{\mid }s\mathrm{\mid }\le 27$ then $G_q$ is non-free, with the possible exception of $s=24$. In this latter case, we prove that the set of denominators $r\in \mathbb{N}$ for which $G_{24\mathrm{/}r}$ is non-free has natural density $1$. For a general numerator $s>27$, we prove that the lower density of denominators $r\in \mathbb{N}$ for which $G_{s\mathrm{/}r}$ is non-free has a lower bound $$1-(1-\frac{11}{s})\prod _{n=1}^{\mathrm{\infty }}(1-\frac{4}{s^{2^n-1}})\mathrm{.}$$ Finally, we show that for a fixed $s$, there are arbitrarily long sequences of consecutive denominators $r$ such that $G_{s\mathrm{/}r}$ is non-free. The proofs of some of the results are computer assisted, and Mathematica code has been provided together with suitable documentation.