Let $A = \pmatrix{1 & 0 \\ \alpha & 1} $ and $ B = \pmatrix{1 & 1 \\ 0 & 1}$, where $\alpha \in \mathbb{C}$ is a complex parameter.

Now consider the family of representations $r_{\alpha}$ of the free group on two generators $F_2 = \langle a,b\rangle$ in $\mathrm{SL}(2, \mathbb{C})$ setting $r_{\alpha}(a) = A$ and $r_{\alpha}(b) = B$. One can see that when $\alpha$ is transcendental over $\mathbb{Q}$, the representation $r_{\alpha}$ is faithful (see T. Church & A. Pixton "Separating twists and the Magnus representation of the Torelli group" Lemma 5.1).

The question I am interested in is the following : when is (or is not) $r_{\alpha}(F_2)$ a discrete subgroup of $\mathrm{SL}(2, \mathbb{C})$ ?

I suppose this is a difficult question of dynamics, I am curious if anyone has ever studied similar questions.