Cohomogeneity one actions on the three-dimensional Einstein universe

TL;DR

This paper classifies all cohomogeneity one conformal group actions on the 3D Einstein universe, identifying reducible and irreducible cases, and describes the structure of their orbits.

Contribution

It provides a complete classification of cohomogeneity one conformal actions on the 3D Einstein universe, including reducible and irreducible subgroups, and analyzes their orbit structures.

Findings

Irreducible action of PSL(2,R) is of cohomogeneity one.

All reducible cohomogeneity one actions induce fixed points in projective space.

Descriptions of all codimension one orbits for these actions.

Abstract

The aim of this paper is to classify the cohomogeneity one conformal actions on the 3-dimensional Einstein universe $Ein^{1;2}$ , up to orbit equivalence. In a recent paper [21], we studied the unique (up to conjugacy) irreducible action of $PSL(2; \mathbb{R})$ on $Ein^{1;2}$ and we showed that the action is of cohomogeneity one. In the present paper, we determine all the subgroups of $Conf(Ein^{1;2})$, up to conjugacy, acting reducibly and with cohomogeneity one on $Ein^{1;2}$. We show that any cohomogeneity one reducible action on $Ein^{1;2}$ 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