Proper affine deformations of positive representations

Bieberbach's theorem states that a compact Euclidean manifold is virtually a torus. What happens if we relax the setting and ask which kinds of manifolds can arise as quotients of affine space by affine group actions? Margulis first found free groups acting affinely with manifold quotient on R^3, and the picture in three dimensions is now well-understood (a satisfying picture is given by Danciger-Guéritaud-Kassel), but remains more elusive in higher dimensions. We will discuss a small piece of the puzzle: given a positive representation of a free group in SO(2n,2n-1), we construct a large set of cocycles twisted by the representation that determine proper actions of the free group by affine transformations on R^(4n-1). We also describe fundamental domains for these actions, bounded by higher-dimensional versions of Drumm's crooked planes. This is joint work with Jean-Philippe Burelle.

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