Simple-stable representations of surface groups in $\mathrm{P}\mathrm{U}(2,1)$

Let ${\mathrm{\Gamma }}_g$ be the fundamental group of a closed orientable surface of genus $g⩾2$. The outer automorphism group $\mathrm{O}\mathrm{u}\mathrm{t}({\mathrm{\Gamma }}_g)$ naturally acts on the character variety $\mathcal{X}({\mathrm{\Gamma }}_g,G)$ for any Lie group $G$. We consider the set of simple-stable representations which are modelled on Minsky’s primitive-stable representations. We prove that the set of conjugacy classes of simple-stable representations of ${\mathrm{\Gamma }}_g$ in $\mathrm{P}\mathrm{U}(2,1)$ is a domain of discontinuity for this action, strictly larger than the set of conjugacy classes of convex cocompact representations.