The combinatorial geometry of Q-Gorenstein quasi-homogeneous surface singularities

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 Γ1 and a cyclic discrete subgroup Γ2 in G˜ which satisfy certain conditions. We describe the Lorentz space form Γ1∖G˜/Γ2 by constructing a fundamental domain for the action of Γ1×Γ2 on G˜ by (g,h)⋅x=gxh−1. This fundamental domain is a polyhedron in the Lorentz manifold G˜ with totally geodesic faces. For a co-compact subgroup Γ1 the corresponding fundamental domain is compact.