
Mathematics 2010
The Combinatorial Geometry of QGorenstein QuasiHomogeneous Surface SingularitiesAbstract: 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 orientationpreserving isometries of the hyperbolic plane. The Killing form on the Lie group G~ gives rise to a biinvariant 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 cocompact 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 biquotients of the form Gamma_1\G~/Gamma_2 are diffeomorphic to the links of quasihomogeneous QGorenstein surface singularities.
