Classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms (Q2418808)

If $G$ is a Lie group, $H$ a closed subgroup of $G$, $\Gamma$ a discrete subgroup of $G$, and $\Gamma$ acts on a homogeneous space $G/H$ properly, discontinuously, and freely, then the double coset space $\Gamma\backslash G/H$ has a natural manifold structure. The double coset space $\Gamma\backslash G/H$ with this manifold structure is called a Clifford-Klein form of $G/H$ and $\Gamma$ is called a discontinuous group for $G/H$. If $G/H$ is a homogeneous space of reductive type and $\Gamma$ is a discontinuous group for $G/H$, then a Clifford-Klein form $\Gamma\backslash G/H$ is called standard if there exists a reductive subgroup $L$ containing $\Gamma$ and acting on $G/H$ properly. In this paper, the author gives a classification of irreducible symmetric spaces which admit standard compact Clifford-Klein forms. It is shown that if $G$ is a linear noncompact semisimple Lie group and $G/H$ is an irreducible symmetric space, and $G/H$ admits a standard compact Clifford-Klein form, then $G/H$ is locally isomorphic to a Riemannian symmetric space $G/K$, or a group manifold $G'\times G'/\text{diag}G'$, or one of the twelve homogeneous spaces admitting proper and cocompact actions of reductive subgroups $L$.