Deformation of Fuchsian representations and proper affine actions

The main goal of this article is to generalize Mess' work and using results from Labourie–Wentworth, Potrie–Sambarino and Smilga, to show that inside Hitchin representations, infinitesimal deformations along certain directions of Fuchsian representations of a cocompact surface group do not act properly on the corresponding Lie algebra. In the process, we introduce affine versions of cross ratios and triple ratios. We introduce Margulis invariants and relate them with affine cross ratios and infinitesimal Jordan projections. We obtain a general equivalent criterion for existence of proper affine actions in terms of Anosov representations and the structure of the Margulis invariant spectra. Also, using a stability argument we show the existence of proper affine actions of non‐abelian free groups whose linear part is a Hitchin representation.