Affine Anosov representations and proper actions

We define the notion of affine Anosov representations of word hyperbolic groups into the affine group ${\mathsf{S}\mathsf{O}}^0(n+1,n)⋉{\mathbb{R}}^{2n+1}$. We then show that a representation $\rho$ of a word hyperbolic group is affine Anosov if and only if its linear part ${\mathtt{L}}_{\rho }$ is Anosov in ${\mathsf{S}\mathsf{O}}^0(n+1,n)$ with respect to the stabilizer of a maximal isotropic plane and $\rho (\mathrm{\Gamma })$ acts properly on ${\mathbb{R}}^{2n+1}$.