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The main result of this paper is a construction of fundamental domains forcertain group actions on Lorentz manifolds of constant curvature. We considerthe simply connected Lie group G~, the universal cover of the group SU(1,1) oforientation-preserving isometries of the hyperbolic plane. The Killing form onthe Lie group G~ gives rise to a bi-invariant Lorentz metric of constantcurvature. We consider a discrete subgroup Gamma_1 and a cyclic discretesubgroup Gamma_2 in G~ which satisfy certain conditions. We describe theLorentz space form Gamma_1\G~/Gamma_2 by constructing a fundamental domain forthe 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 totallygeodesic faces. For a co-compact subgroup the corresponding fundamental domainis compact. The class of subgroups for which we construct fundamental domainscorresponds to an interesting class of singularities. The bi-quotients of theform Gamma_1\G~/Gamma_2 are diffeomorphic to the links of quasi-homogeneousQ-Gorenstein surface singularities.

The Combinatorial Geometry of Q-Gorenstein Quasi-Homogeneous Surface Singularities