Totally elliptic surface group representations

A surface group representation into a Lie group is called totally elliptic if every simple closed curve on the surface is mapped to an elliptic element of the target group. In this note, we characterize all totally elliptic surface group representations into ${\mathrm{P}\mathrm{S}\mathrm{L}}_2\mathbb{R}$ and ${\mathrm{P}\mathrm{S}\mathrm{L}}_2\mathbb{C}$ by showing that they are either representations into a compact subgroup or Deroin–Tholozan representations.