Isometries of Lorentz surfaces and convergence groups

We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of $\mathrm{Diff}(\mathbb{S}^1)$ obtained are semi conjugate to subgroups of finite covers of $\mathrm{PSL}(2,\mathbb{R})$ by using convergence groups. Under an assumption on the conformal boundary, we show that we have a conjugacy in $\mathrm{Homeo}(\mathbb{S}^1)$.