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where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on the homogeneous space G/H. A suitable discrete subgroup Γ may or may not exist, for a given G and H. If Γ exists, there is the question of whether Γ\G/H can be taken to be a compact space, called a compact Clifford–Klein form.
When H is itself compact, classical results show that a compact Clifford–Klein form exists. Otherwise it may not, and there are a number of negative results.