Abstract
A homogeneous space G/H is said to have a compact Clifford–Klein form if there exists a discrete subgroup Γ of G that acts properly discontinuously on G/H, such that the quotient space Γ\G/H is compact. When n is even, we find every closed, connected subgroup H of G = SO(2, n), such that G/H has a compact Clifford–Klein form, but our classification is not quite complete when n is odd. The work reveals new examples of homogeneous spaces of SO(2, n) that have compact Clifford–Klein forms, if n is even. Furthermore, we show that if H is a closed, connected subgroup of G = SL(3, R), and neither H nor G/H is compact, then G/H does not have a compact Clifford–Klein form, and we also study noncompact Clifford–Klein forms of finite volume.
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Oh, H., Witte, D. Compact Clifford–Klein Forms of Homogeneous Spaces of SO(2, n). Geometriae Dedicata 89, 25–56 (2002). https://doi.org/10.1023/A:1014227302011
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DOI: https://doi.org/10.1023/A:1014227302011