On rank two Kleinian groups with three parabolics

The free group of rank $2$ is the fundamental group of the genus $2$ handlebody $\mathcal{H}$.

We study discrete representations of this group into $ \mathsf{PSL}(2,\mathbb{C}) $ so that three disjoint simple closed curves on the conformal boundary $\partial_\infty \mathcal{H} $ are sent to parabolic elements.

We show that the only infinite covolume groups of this form are maximal cusp groups on the boundary of genus $2$ Schottky space.

We also exhibit hitherto unexpected finite covolume groups which do not arise from Heegaard splitting presentations of tunnel number $1$ links.