Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous spaces, namely quotients by discrete groups Gamma acting properly discontinuously, in the case that Gamma is word hyperbolic and acts via an Anosov representation. In particular, these Clifford-Klein forms are topologically tame.