Fundamental Domains in Lorentzian Geometry

Abstract

We consider discrete subgroups Gamma of the simply connected Lie group SU~(1,1), the universal cover of SU(1,1), of finite level, i.e. the subgroup intersects the centre of SU~(1,1) in a subgroup of finite index, this index is called the level of the group. The Killing form induces a Lorentzian metric of constant curvature on the Lie group SU~(1,1). The discrete subgroup Gamma acts on SU~(1,1) by left translations. We describe the Lorentz space form SU~(1,1)/Gamma by constructing a fundamental domain F for Gamma. We want F to be a polyhedron with totally geodesic faces. We construct such F for all Gamma satisfying the following condition: The image of Gamma in PSU(1,1) has a fixed point u in the unit disk of order larger than the index of Gamma. The construction depends on the group Gamma and on the orbit Gamma(u) of the fixed point u.