The combinatorial geometry of [formula omitted]-Gorenstein quasi-homogeneous surface singularities

Abstract

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G ˜ = SU ˜ ( 1 , 1 ) . The Killing form on the Lie group G ˜ gives rise to a bi-invariant Lorentz metric of constant curvature. We consider a discrete subgroup Γ 1 and a cyclic discrete subgroup Γ 2 in G ˜ which satisfy certain conditions. We describe the Lorentz space form Γ 1 ∖ G ˜ / Γ 2 by constructing a fundamental domain for the action of Γ 1 × Γ 2 on G ˜ by ( g , h ) ⋅ x = g x h − 1 . This fundamental domain is a polyhedron in the Lorentz manifold G ˜ with totally geodesic faces. For a co-compact subgroup Γ 1 the corresponding fundamental domain is compact.