New examples of compact Clifford-Klein forms of homogeneous spaces of SO(2, n)

Abstract

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 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. We also study noncompact Clifford-Klein forms of finite volume.