Cohomogeneity one three dimensional anti-de Sitter space, proper and nonproper actions

In this paper we give a classification of closed and connected Lie groups, up to conjugacy in Iso(adS3), acting by cohomogeneity one on the three dimensional anti-de Sitter space adS3. Then we determine the causal character of the orbits and then the orbit spaces, up to homeomorphism, in both cases, proper and nonproper actions. When the action is proper, we show that there is no exceptional orbit and causal characters of the principal orbits are the same.