  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.
