I am planning to focus on representations $\rho:\langle a,b\rangle\to \operatorname{PSL}_2(\mathbb{C})$ of the free group on 2 generators such that the image $\rho(a^{-1}b^{-1}ab)$ is parabolic, and the corresponding character variety $X$.

What subset of $X$ corresponds to representations with discrete image? The discrete faithful representations naturally form a product of two Teichmüller spaces, in this case of two discs. I will describe this isomorphism, and how it helps to understand the discrete faithful representations, especially its boundary.