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 Diff(S^1) obtained are semi conjugate to subgroups of finite covers of PSL(2,R) by using convergence groups. Under an assumption on the conformal boundary, we show that we have a conjugacy in Homeo(S^1 )