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

Let $\Gamma_g$ be the fundamental group of a closed orientable surface of genus $g\geqslant 2$. The outer automorphism group $\mathrm{Out}(\Gamma_g)$ naturally acts on the character variety $\mathcal{X}(\Gamma_g,G)$ 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 $\Gamma_g$ in $\mathrm{PU}(2,1)$ is a domain of discontinuity for this action, strictly larger than the set of conjugacy classes of convex cocompact representations.