Proper affine deformations of positive representations

We define for every positive Anosov representation of a nonabelian free group into $\mathrm{S}\mathrm{O}(2n,2n-1)$ a family of ${\mathbb{R}}^{4n-1}$-valued cocycles which induce proper affine actions on ${\mathbb{R}}^{4n-1}$. We construct fundamental domains in ${\mathbb{R}}^{4n-1}$ bounded by generalized crooked planes for these affine actions, and deduce that the quotient manifolds are homeomorphic to handlebodies.