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~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie group G~ gives rise to a bi-invariant Lorentz metric of constant curvature. We consider a discrete subgroup Gamma_1 and a cyclic discrete subgroup Gamma_2 in G~ which satisfy certain conditions. We describe the Lorentz space form Gamma_1\G~/Gamma_2 by constructing a fundamental domain for the action of the product of Gamma_1 and Gamma_2 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 the corresponding fundamental domain is compact. The class of subgroups for which we construct fundamental domains corresponds to an interesting class of singularities. The bi-quotients of the form Gamma_1\G~/Gamma_2 are diffeomorphic to the links of quasi-homogeneous Q-Gorenstein surface singularities.