Lyndon–Ullman–Kim–Koberda conjecture on freeness of parabolic matrix groups

Let $q\in\mathbb{C}$ and let $\Delta_q$ be the subgroup of $\operatorname{SL}(2,\mathbb{C})$ generated by

$$a=\begin{pmatrix}1&0\\1&1\end{pmatrix},\qquad b_q=\begin{pmatrix}1&q\\0&1\end{pmatrix}.$$

Lyndon–Ullman–Kim–Koberda conjecture. For every nonzero rational value of $q$ in $(-4,4)$, the group $\Delta_q$ is not free.

Freeness for these nonzero rational parameters is described as a long-standing open problem, following work of Lyndon–Ullman and a negative conjecture attributed to Kim–Koberda. The claim concerns the algebraic structure of groups generated by parabolic elements; Sanov's theorem proves freeness for $q=4$, while Brenner proved freeness and discreteness for $|q|>4$.