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• W39160967 abstract "Let G be a Lie group and H a subgroup. A Clifford-Klein form of the homogeneous manifold G/H is a double coset space / G/H , where is a subgroup of G acting properly discontinuously and freely on G/H . For example, any closed Riemann surface M with genus ≥2 is biholomorphic to a compact Clifford-Klein form of the Poincare plane G/H =PSL(2,ℝ葷)/SO(2). On the other hand, there is no compact Clifford-Klein form of the hyperboloid of one sheet G/H =PSL(2,ℝ葷)/SO(1,1). Even more, there is no infinite discrete subgroup of G that acts properly discontinuously on G/H (the Calabi-Markus phenomenon). We discuss recent developments in the theory of discontinuous groups acting on G/H where G is a real reductive Lie group and H a noncompact reductive subgroup. Geometric ideas of various methods together with a number of examples are presented regarding the fundamental problems: Which homogeneous manifolds G/H admit properly discontinuous actions of infinite discrete subgroups of G? Which homogenous manifolds admit compact Clifford-Klein forms?" @default.
• W39160967 created "2016-06-24" @default.
• W39160967 creator A5041756388 @default.
• W39160967 date "1997-01-01" @default.
• W39160967 modified "2023-10-01" @default.
• W39160967 title "Discontinuous Groups and Clifford—Klein Forms of Pseudo-Riemannian Homogeneous Manifolds" @default.
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• W39160967 doi "https://doi.org/10.1016/b978-012625440-2/50004-5" @default.
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