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

Pratoussevitch, Anna ORCID: 0000-0003-2248-6382
The Combinatorial Geometry of Q-Gorenstein Quasi-Homogeneous Surface Singularities. Differential Geometry and its Applications 29 (2011), 507-515.

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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.

Item Type: Article
Uncontrolled Keywords: math.DG, math.DG, Primary 53C50, Secondary 14J17, 32S25, 51M20, 52B10
Depositing User: Symplectic Admin
Date Deposited: 11 Apr 2016 13:44
Last Modified: 12 Nov 2019 14:35
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