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 de
duce 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.
We establish several characterizations of Anosov representations of word hyperbolic groups into real reductive Lie groups, in terms of a Cartan projection or Lyapunov projection of the Lie group. Using a properness criterion of Benoist and Kobayashi,
we derive applications to proper actions on homogeneous spaces of reductive groups.
