Cohomogeneity one actions on the three-dimensional Einstein universe

Abstract

The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe $Ei{n}^{1;2}$ , up to orbit equivalence. In a recent paper [21], we studied the unique (up to conjugacy) irreducible action of $PSL(2;\mathbb{R})$ on $Ei{n}^{1;2}$ and we showed that the action is of cohomogeneity one. In the present paper, we determine all the subgroups of $Conf(Ei{n}^{1;2})$, up to conjugacy, acting reducibly and with cohomogeneity one on $Ei{n}^{1;2}$. We show that any cohomogeneity one reducible action on $Ei{n}^{1;2}$ induces a fixed point in the 4-dimensional projective space $\mathbb{R}{\mathbb{P}}^4$. Also, we describe all the codimension one induced orbits by these actions